text: 2 MARK ALLEN (cid:90) (cid:90) (1.3) u v + aij(x, t)u v = fv. t i j Ω Ω Solutions to (1.3) are considered weak solutions that are strong in time. If u is a solution to (1.3), then by integrating in time and transferring the derivative in time to the(cid:90) test function, u is(cid:90)also(cid:90) a solution to (cid:90) (cid:90) − T T (1.4) u(x, 0)v(x, 0) uv + aij(x, t)u v = f(x, t)v(x, t) t i j Ω 0 Ω 0 Ω ∈ (cid:48) ∈ ≡ for all v L2(0, T; H1(Ω)) with v L2(0, T; L2(Ω)) and v(x, T) 0. Proving 0 uniqueness for solutions to (1.3) is straightforward since u is a valid test function, so one simply multiplies by u and integrates in time. Proving uniqueness for solutions to (1.4) is not straightforward, however, since a priori it is not known if u is a valid test function. In this paper we say “weak-in-time” to emphasize that via an integration by parts formula, the fractional time derivative is transferred to the test function as in (1.4). It is advantageous to consider solutions to (1.4) which are weak solutions in time since it is often convenient [8] to work in the larger class of solutions to prove existence as well as regularity estimates such as H¨older continuity. This is also true for parabolic equations involving the Caputo derivative. Zacher [12] utilized a larger class of solutions to prove H¨older continuity of solutions to parabolic equations involving the Caputo derivative which are local in the spatial variables. Recently in [3] the author with Caﬀarelli and Vasseur proved H¨older continuity for solutions to a nonlocal parabolic equation of divergence form which is a nonlocal analogue (cid:90) to (1.2) and is given by − − (1.5) ∂αu [w(y, t) w(x, t)]K(x, y, t) = f(x, t). t The authors in [3] utilized solutions that are weak-in-time to prove H¨older conti- nuity. The same authors also adapted the methods in [4] to prove H¨older continuity for a nonlocal porous medium equation. We also remark that H¨older continuity for nondivergence form parabolic equations involving fractional derivatives was shown in [2] and [1]. Considering weak-in-time solutions is advantageous for existence and regularity results; however, proving uniqueness becomes more diﬃcult. Uniqueness for weak- in-time solutions to (1.5) was only shown in [3] if K(x, y, t) is independent of t. In this paper we show uniqueness for K(x, y, t) with no regularity assumptions on the variable t. As the methods apply to the more general parabolic equation (1.1), we show uniqueness for weak-in-time solutions to (1.1). In order to formulate weak-in-time solutions, we recall the deﬁnition of the Caputo derivative and an integration by parts formula. The Caputo fractional time derivative fo(cid:90)r 0 < α < 1 is deﬁned as (cid:48) 1 t u (s) (1.6) ∂αu(t) := − − ds, a t Γ(1 α) (t s)α a ##LTLine## ##LTLine##

label: paragraph

text: MARK

text: ALLEN

label: equation

text: u

text: v

text: aij(x,

text: t)u

text: v

text: fv.

text: t

text: i

text: j

text: Ω

text: Ω

text: Solutions

text: to

text: are

text: considered

text: weak

text: solutions

text: that

text: are

text: strong

text: in

text: time.

text: If

text: u

text: is

text: a

text: solution

text: to

text: then

text: by

text: integrating

text: in

text: time

text: and

text: transferring

text: the

text: derivative

text: in

text: time

text: to

text: test

text: function,

text: u

text: a

text: solution

text: to

text: T

text: T

text: u(x,

text: uv

text: aij(x,

text: t)u

text: v

text: f(x,

text: t)v(x,

text: t)

text: t

text: i

text: j

text: Ω

text: Ω

text: Ω

text: for

text: all

text: v

text: T;

text: with

text: v

text: T;

text: and

text: v(x,

text: T)

text: Proving

text: uniqueness

text: for

text: solutions

text: to

text: is

text: straightforward

text: since

text: u

text: is

text: a

text: valid

text: test

text: function,

text: so

text: one

text: simply

text: multiplies

text: by

text: u

text: and

text: integrates

text: in

text: time.

text: Proving

text: uniqueness

text: for

text: solutions

text: to

text: is

text: not

text: straightforward,

text: however,

text: since

text: a

text: priori

text: it

text: is

text: not

text: known

text: if

text: u

text: is

text: a

text: valid

text: test

text: function.

text: In

text: this

text: paper

text: we

text: say

text: “weak-in-time”

text: to

text: emphasize

text: that

text: via

text: an

text: integration

text: by

text: parts

text: formula,

text: the

text: fractional

text: time

text: derivative

text: is

text: transferred

text: to

text: the

text: test

text: function

text: as

text: in

text: It

text: is

text: advantageous

text: to

text: consider

text: solutions

text: to

text: which

text: are

text: weak

text: solutions

text: in

text: time

text: since

text: it

text: is

text: often

text: convenient

text: to

text: work

text: in

text: the

text: larger

text: class

text: of

text: solutions

text: to

text: prove

text: existence

text: as

text: well

text: as

text: regularity

text: estimates

text: such

text: as

text: H¨older

text: continuity.

text: This

text: is

text: also

text: true

text: for

text: parabolic

text: equations

text: involving

text: the

text: Caputo

text: derivative.

text: Zacher

text: utilized

text: a

text: larger

text: class

text: of

text: solutions

text: to

text: prove

text: H¨older

text: continuity

text: of

text: solutions

text: to

text: parabolic

text: equations

text: involving

text: the

text: Caputo

text: derivative

text: which

text: are

text: local

text: in

text: the

text: spatial

text: variables.

text: Recently

text: in

text: the

text: author

text: with

text: Caﬀarelli

text: and

text: Vasseur

text: proved

text: H¨older

text: continuity

text: for

text: solutions

text: to

text: a

text: nonlocal

text: parabolic

text: equation

text: of

text: divergence

text: form

text: which

text: is

text: a

text: nonlocal

text: analogue

text: to

text: and

text: is

text: given

text: by

text: ∂αu

text: [w(y,

text: t)

text: w(x,

text: t)]K(x,

text: y,

text: t)

text: f(x,

text: t).

text: t

text: The

text: authors

text: in

text: utilized

text: solutions

text: that

text: are

text: weak-in-time

text: to

text: prove

text: H¨older

text: conti-

text: nuity.

text: The

text: same

text: authors

text: also

text: adapted

text: the

text: methods

text: in

text: to

text: prove

text: H¨older

text: continuity

text: for

text: a

text: nonlocal

text: porous

text: medium

text: equation.

text: We

text: also

text: remark

text: that

text: H¨older

text: continuity

text: for

text: nondivergence

text: form

text: parabolic

text: equations

text: involving

text: fractional

text: derivatives

text: was

text: shown

text: in

text: and

text: Considering

text: weak-in-time

text: solutions

text: is

text: advantageous

text: for

text: existence

text: and

text: regularity

text: results;

text: however,

text: proving

text: uniqueness

text: becomes

text: more

text: diﬃcult.

text: Uniqueness

text: for

text: weak-

text: in-time

text: solutions

text: to

text: was

text: only

text: shown

text: in

text: if

text: K(x,

text: y,

text: t)

text: is

text: independent

text: of

text: t.

text: In

text: this

text: paper

text: we

text: show

text: uniqueness

text: for

text: K(x,

text: y,

text: t)

text: with

text: no

text: regularity

text: assumptions

text: on

text: the

text: variable

text: t.

text: As

text: the

text: methods

text: apply

text: to

text: the

text: more

text: general

text: parabolic

text: equation

text: we

text: show

text: uniqueness

text: for

text: weak-in-time

text: solutions

text: to

text: In

text: order

text: to

text: formulate

text: weak-in-time

text: solutions,

text: we

text: recall

text: the

text: deﬁnition

text: of

text: the

text: Caputo

text: derivative

text: and

text: an

text: integration

text: by

text: parts

text: formula.

text: The

text: Caputo

text: fractional

text: time

text: derivative

text: α

text: is

text: deﬁned

text: as

text: t

text: u

text: (s)

text: ∂αu(t)

text: ds,

text: a

text: t

text: α)

text: (t

text: s)α

text: a

text: ##LTLine##

text: ##LTLine##


text: UNIQUENESS FOR WEAK SOLUTIONS 5 (cid:90) (cid:90) (cid:90) − − T ((cid:90)u(t), φ(t))H T (cid:90) t (u(t) u(s), φ(t) φ(s))H − dt + α − ds dt (T t)1+α −∞ (t s)1+α 0 0 − (cid:90) T (cid:20) T (cid:21) (u(t), ∂tαφ(t))H dt + a(t, u(t), φ(t)) dt (1.14) 0 0 (cid:90) 0 1 − 1 + (v(t), φ(t))H − − dt −∞ (T t)α (0 t)α T (cid:104) (cid:105) = f, φ V (cid:48)×V dt 0 for all ∈ φ L2(0, T; V ) ∈ ∂αφ L2(0, T; H ) t − − ∈ −∞ φ(t)(1 t) (1+α)/2 L2( , 0; H ) ≡ ≤ − φ(t) 0 for t M for some M > 0. ≡ ≤ − Since for any test function φ, we assume φ(t) 0 for t M for some M > 0, ∈ − and since φ L2( M, T; H ), from [5] the Marchaud derivative of φ can be deﬁned in the distributional sense, and this is how to understand the second requirement above. For the test function we will use later, ∂αφ will be deﬁned classically every- t where and will be continuous. Our main theorem is the following uniqueness result. Theorem 1.1. Solutions to (1.14) are unique; i.e. if u , u are both solutions 1 2 − ≡ to (1.14) with prescribed initial data v(t) and right hand side f(t), then u u 0. 1 2 A future question of interest would be to determine necessary and suﬃcient conditions on the initial data v(t), the bilinear form a, and right hand side f, ∈ so that for a solution u of (1.14), the fractional derivative ∂αu L2(0, T; H ) and t consequently u is a strong solution to (1.14). Another question of interest is proving uniqueness for weak-in-time solutions for fractional derivatives deﬁned by a kernel ≈ − − − − K(t, s) (t s) 1 α and satisfying K(t, t s) = K(t + s, t). Such kernels were considered in [4] and the H¨older continuity results in [3] will also apply to fractional derivatives deﬁned by such a kernel. As this paper is concerned with uniqueness, we do not prove existence of so- lutions to (1.14). However, we do mention that for a speciﬁc V and H and a restricted class of right-hand side functions, existence was proven in [3]. To prove uniqueness we use Steklov averages. In Section we present preliminary results on how the Steklov averages behave with the fractional Marchaud derivative. In Section we prove our main result. Rather than show that any solution u to (1.14) is contained in successively better spaces as is done in the local case (see [8]), we utilize the nonlocal nature of the Marchaud derivative to prove uniqueness directly. This is the essence of Lemma 3.1. We deﬁne the notation that will be consistent throughout the paper. • ∂α - the Marchaud derivative as deﬁned in (1.9). t ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: UNIQUENESS

text: FOR

text: WEAK

text: SOLUTIONS

text: T

text: φ(t))H

text: T

text: t

text: (u(t)

text: u(s),

text: φ(t)

text: φ(s))H

text: dt

text: α

text: ds

text: dt

text: (T

text: (t

text: T

text: T

text: (u(t),

text: ∂tαφ(t))H

text: dt

text: a(t,

text: u(t),

text: φ(t))

text: dt

text: (v(t),

text: φ(t))H

text: dt

text: (T

text: t)α

text: t)α

text: T

text: f,

text: φ

text: V

text: dt

text: for

text: all

text: φ

text: T;

text: V

text: ∂αφ

text: T;

text: H

text: t

text: t)

text: H

text: φ(t)

text: for

text: t

text: M

text: for

text: some

text: M

text: Since

text: for

text: any

text: test

text: function

text: φ,

text: we

text: assume

text: φ(t)

text: for

text: t

text: M

text: for

text: some

text: M

text: and

text: since

text: φ

text: M,

text: T;

text: H

text: from

text: the

text: Marchaud

text: derivative

text: of

text: φ

text: can

text: be

text: deﬁned

text: in

text: the

text: distributional

text: sense,

text: and

text: this

text: is

text: how

text: to

text: understand

text: the

text: second

text: requirement

text: above.

text: For

text: the

text: test

text: function

text: we

text: will

text: use

text: later,

text: ∂αφ

text: will

text: be

text: deﬁned

text: classically

text: every-

text: t

text: where

text: and

text: will

text: be

text: continuous.

text: Our

text: main

text: theorem

text: is

text: the

text: following

text: uniqueness

text: result.

text: Theorem

text: Solutions

text: to

text: are

text: unique;

text: i.e.

text: if

text: u

text: u

text: are

text: both

text: solutions

text: to

text: with

text: prescribed

text: initial

text: data

text: v(t)

text: and

text: right

text: hand

text: side

text: f(t),

text: then

text: u

text: u

text: A

text: future

text: question

text: of

text: interest

text: would

text: be

text: to

text: determine

text: necessary

text: and

text: suﬃcient

text: conditions

text: on

text: the

text: initial

text: data

text: v(t),

text: the

text: bilinear

text: form

text: a,

text: and

text: right

text: hand

text: side

text: f,

text: so

text: that

text: for

text: a

text: solution

text: u

text: of

text: the

text: fractional

text: derivative

text: ∂αu

text: T;

text: H

text: and

text: t

text: consequently

text: u

text: is

text: a

text: strong

text: solution

text: to

text: Another

text: question

text: of

text: interest

text: is

text: proving

text: uniqueness

text: for

text: weak-in-time

text: solutions

text: for

text: fractional

text: derivatives

text: deﬁned

text: by

text: a

text: kernel

text: K(t,

text: s)

text: (t

text: s)

text: α

text: and

text: satisfying

text: K(t,

text: t

text: s)

text: K(t

text: s,

text: t).

text: Such

text: kernels

text: were

text: considered

text: in

text: and

text: the

text: H¨older

text: continuity

text: results

text: in

text: will

text: also

text: apply

text: to

text: fractional

text: derivatives

text: deﬁned

text: by

text: such

text: a

text: kernel.

text: As

text: this

text: paper

text: is

text: concerned

text: with

text: uniqueness,

text: we

text: do

text: not

text: prove

text: existence

text: of

text: so-

text: lutions

text: to

text: However,

text: we

text: do

text: mention

text: that

text: for

text: a

text: speciﬁc

text: V

text: and

text: H

text: and

text: a

text: restricted

text: class

text: of

text: right-hand

text: side

text: functions,

text: existence

text: was

text: proven

text: in

text: To

text: prove

text: uniqueness

text: we

text: use

text: Steklov

text: averages.

text: In

text: Section

text: we

text: present

text: preliminary

text: results

text: on

text: how

text: the

text: Steklov

text: averages

text: behave

text: with

text: the

text: fractional

text: Marchaud

text: derivative.

text: In

text: Section

text: we

text: prove

text: our

text: main

text: result.

text: Rather

text: than

text: show

text: that

text: any

text: solution

text: u

text: to

text: is

text: contained

text: in

text: successively

text: better

text: spaces

text: as

text: is

text: done

text: in

text: the

text: local

text: case

text: (see

text: we

text: utilize

text: the

text: nonlocal

text: nature

text: of

text: the

text: Marchaud

text: derivative

text: to

text: prove

text: uniqueness

text: directly.

text: This

text: is

text: the

text: essence

text: of

text: Lemma

text: We

text: deﬁne

text: the

text: notation

text: that

text: will

text: be

text: consistent

text: throughout

text: the

text: paper.

text: ∂α

text: the

text: Marchaud

text: derivative

text: as

text: deﬁned

text: in

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: and Shishikura who constructed a wandering domain of connectivity two by using quasi-conformal surgery, see [53] for details. P II. Wandering domains for functions in class . Baker [8], Kotus [55] and Mukhamedshin [65] showed that there are functions in P class having wandering domains by using approximation theory. Mart´ı-Pete gave in [59] an explicit example, f(z) = ze(sinz+2π)/z which has a bounded wandering domain U, the orbit of points in U tend to inﬁnity. M III. Wandering domains for functions in class . In [11] the authors presented, using results on complex approximation, examples of wandering domains (bounded or unbounded) of any connectivity, so they solved the connectivity problem for transcendental meromorphic functions. K IV. Wandering domains for functions in class . K Doubly connected wandering domains for funtions in class have been given in [36], by using quasi-conformal surgery. In Section , of this paper, we give an example of a ∈ K function f which has inﬁnity wandering domains with disjoint orbits. K 5 Some results related to functions in class For a periodic component in the Fatou set, Bolsch in [26] prove the following theorem by using Ahlfors theory of covering surfaces. ∈ K Theorem 5.1. If f and U is a periodic component of the Fatou set, then U has ∞ connectivity 1, 2 or . The above result improves the result of Baker, Kotus and Lu¨ given in [12] for invariant components. Concerning completely invariant components in the Fatou set the Lemmas 4.1, 4.2 K and 4.3 in [12] can be generalize to functions in class . ∈ K Lemma 5.2. Let f . A completely invariant component of the Fatou set U has ∞ connectivity either 1 or . ∈ K Lemma 5.3. Let f and U be a completely invariant component of the Fatou set. Then ∂U = J(f). ∈ K Lemma 5.4. Let f . If there are two or more completely invariant components in the Fatou set, then each one is simply connected. As an analogy with the deﬁnition given by Eremenko and Lyubich in [40] we deﬁne S ∈ K the class K as the set of functions f such that the set of singular values is ﬁnite. 16

text: and

text: Shishikura

text: who

text: constructed

text: a

text: wandering

text: domain

text: of

text: connectivity

text: two

text: by

text: using

text: quasi-conformal

text: surgery,

text: see

text: for

text: details.

text: P

text: II.

text: Wandering

text: domains

text: for

text: functions

text: in

text: class

text: Baker

text: Kotus

text: and

text: Mukhamedshin

text: showed

text: that

text: there

text: are

text: functions

text: in

text: P

text: class

text: having

text: wandering

text: domains

text: by

text: using

text: approximation

text: theory.

text: Mart´ı-Pete

text: gave

text: in

text: an

text: explicit

text: example,

text: f(z)

text: which

text: has

text: a

text: bounded

text: wandering

text: domain

text: U,

text: the

text: orbit

text: of

text: points

text: in

text: U

text: tend

text: to

text: inﬁnity.

text: M

text: III.

text: Wandering

text: domains

text: for

text: functions

text: in

text: class

text: In

text: the

text: authors

text: presented,

text: using

text: results

text: on

text: complex

text: approximation,

text: examples

text: of

text: wandering

text: domains

text: (bounded

text: or

text: unbounded)

text: of

text: any

text: connectivity,

text: so

text: they

text: solved

text: the

text: connectivity

text: problem

text: for

text: transcendental

text: meromorphic

text: functions.

text: K

text: IV.

text: Wandering

text: domains

text: for

text: functions

text: in

text: class

text: K

text: Doubly

text: connected

text: wandering

text: domains

text: for

text: funtions

text: in

text: class

text: have

text: been

text: given

text: in

text: by

text: using

text: quasi-conformal

text: surgery.

text: In

text: Section

text: of

text: this

text: paper,

text: we

text: give

text: an

text: example

text: of

text: a

text: K

text: function

text: f

text: which

text: has

text: inﬁnity

text: wandering

text: domains

text: with

text: disjoint

text: orbits.

text: K

text: Some

text: results

text: related

text: to

text: functions

text: in

text: class

text: For

text: a

text: periodic

text: component

text: in

text: the

text: Fatou

text: set,

text: Bolsch

text: in

text: prove

text: the

text: following

text: theorem

text: by

text: using

text: Ahlfors

text: theory

text: of

text: covering

text: surfaces.

text: K

text: Theorem

text: If

text: f

text: and

text: U

text: is

text: a

text: periodic

text: component

text: of

text: the

text: Fatou

text: set,

text: then

text: U

text: has

text: connectivity

text: or

text: The

text: above

text: result

text: improves

text: the

text: result

text: of

text: Baker,

text: Kotus

text: and

text: Lu¨

text: given

text: in

text: for

text: invariant

text: components.

text: Concerning

text: completely

text: invariant

text: components

text: in

text: the

text: Fatou

text: set

text: the

text: Lemmas

text: K

text: and

text: in

text: can

text: be

text: generalize

text: to

text: functions

text: in

text: class

text: K

text: Lemma

text: Let

text: f

text: A

text: completely

text: invariant

text: component

text: of

text: the

text: Fatou

text: set

text: U

text: has

text: connectivity

text: either

text: or

text: K

text: Lemma

text: Let

text: f

text: and

text: U

text: be

text: a

text: completely

text: invariant

text: component

text: of

text: the

text: Fatou

text: set.

text: Then

text: ∂U

text: J(f).

text: K

text: Lemma

text: Let

text: f

text: If

text: there

text: are

text: two

text: or

text: more

text: completely

text: invariant

text: components

text: in

text: the

text: Fatou

text: set,

text: then

text: each

text: one

text: is

text: simply

text: connected.

text: As

text: an

text: analogy

text: with

text: the

text: deﬁnition

text: given

text: by

text: Eremenko

text: and

text: Lyubich

text: in

text: we

text: deﬁne

text: S

text: K

text: the

text: class

text: K

text: as

text: the

text: set

text: of

text: functions

text: f

text: such

text: that

text: the

text: set

text: of

text: singular

text: values

text: is

text: ﬁnite.


text: ∗ Deﬁnition 6.23. Let O (p, γ, f) be a singular orbit of an endpoint p of an escaping hair γ to E, p can be classiﬁed as follows: 1. periodic singular end point if a singular orbit is periodic, 2. pre-periodic singular end point if a singular orbit is pre-periodic, 3. oscillating singular end point if #E > 1 and, 4. wandering singular end point if its singular orbit is an inﬁnite set. Observe that (i) by deﬁnition every pre-periodic singular end point is periodic and (ii) by choosing diﬀerent escaping hairs with a singular end point p, it may have diﬀerent classiﬁcation. ∈ K ∈ − Question 6.24. Is there an example of a function f and p B (f) such that p has singular orbits for each of the cases in the previous deﬁnition? We are deﬁning in some way the dynamics of a singular end point p using escaping ∈ N hairs, even when fn(p) is not deﬁned for all n . In Section we give an explicit K example of a function in class with a wandering singular end point. − ∈ R− −∞ In the example f(z) = ez + 1/z, there is x such that γ = ( , x] is {∞ } ∞ an escaping hair to E = , 0 with singular end point which is periodic, with ∗ ∞ {∞ ∞ } singular orbit O ( , γ, f) = , 0, , 0, ... . While f(γ) is an escaping hair with zero ∗ { ∞ ∞ } as a periodic singular end point and singular orbit O (0, f(γ), f) = 0, , 0, , ... , see Figure 2. Due to a result in [80], I∞(f) contains a escaping hair γ with singular ∞ ∗ ∞ {∞ ∞ ∞ } end point , so its singular obit O ( , γ, f) = , , , ... . Therefore, in this case, ∞ p = has diﬀerent dynamics according to Deﬁnition 6.23. We mention below some open question related to the previous discussion. Question 6.25. Is there an example of a pre-periodic singular end point? Question 6.26. Is there an example of an oscillating singular end point non periodic neither pre-periodic? Question 6.27. Is there an example of an escaping hair to E such that E is inﬁnite? 7 Example of escaping hairs with wandering singu- lar end points In this section we give two examples, in the ﬁrst one we show that for each of the ﬁnite essential singularities of the function g(z) = e 1 + z, has inﬁnitely many Baker sinz domains attached to it. In the second one, for each ﬁnite essential singularity of the function f(z) = e 1 + z + 2π, there are inﬁnitely many wandering domains attached sinz 23 ##LTLine## ##LTLine##

text: Deﬁnition

text: Let

text: O

text: (p,

text: γ,

text: f)

text: be

text: a

text: singular

text: orbit

text: of

text: an

text: endpoint

text: p

text: of

text: an

text: escaping

text: hair

text: γ

text: to

text: E,

text: p

text: can

text: be

text: classiﬁed

text: as

text: follows:

label: list

text: periodic

text: singular

text: end

text: point

text: if

text: a

text: singular

text: orbit

text: is

text: periodic,

text: pre-periodic

text: singular

text: end

text: point

text: if

text: a

text: singular

text: orbit

text: is

text: pre-periodic,

text: oscillating

text: singular

text: end

text: point

text: if

text: #E

text: and,

text: wandering

text: singular

text: end

text: point

text: if

text: its

text: singular

text: orbit

text: is

text: an

text: inﬁnite

text: set.

text: Observe

text: that

text: (i)

text: by

text: deﬁnition

text: every

text: pre-periodic

text: singular

text: end

text: point

text: is

text: periodic

text: and

text: (ii)

text: by

text: choosing

text: diﬀerent

text: escaping

text: hairs

text: with

text: a

text: singular

text: end

text: point

text: p,

text: it

text: may

text: have

text: diﬀerent

text: classiﬁcation.

text: K

text: Question

text: Is

text: there

text: an

text: example

text: of

text: a

text: function

text: f

text: and

text: p

text: B

text: (f)

text: such

text: that

text: p

text: has

text: singular

text: orbits

text: for

text: each

text: of

text: the

text: cases

text: in

text: the

text: previous

text: deﬁnition?

text: We

text: are

text: deﬁning

text: in

text: some

text: way

text: the

text: dynamics

text: of

text: a

text: singular

text: end

text: point

text: p

text: using

text: escaping

text: N

text: hairs,

text: even

text: when

text: fn(p)

text: is

text: not

text: deﬁned

text: for

text: all

text: n

text: In

text: Section

text: we

text: give

text: an

text: explicit

text: K

text: example

text: of

text: a

text: function

text: in

text: class

text: with

text: a

text: wandering

text: singular

text: end

text: point.

text: R−

text: In

text: the

text: example

text: f(z)

text: ez

text: there

text: is

text: x

text: such

text: that

text: γ

text: x]

text: is

text: an

text: escaping

text: hair

text: to

text: E

text: with

text: singular

text: end

text: point

text: which

text: is

text: periodic,

text: with

text: singular

text: orbit

text: O

text: γ,

text: f)

text: While

text: f(γ)

text: is

text: an

text: escaping

text: hair

text: with

text: zero

text: as

text: a

text: periodic

text: singular

text: end

text: point

text: and

text: singular

text: orbit

text: O

text: f(γ),

text: f)

text: see

text: Figure

text: Due

text: to

text: a

text: result

text: in

text: I∞(f)

text: contains

text: a

text: escaping

text: hair

text: γ

text: with

text: singular

text: end

text: point

text: so

text: its

text: singular

text: obit

text: O

text: γ,

text: f)

text: Therefore,

text: in

text: this

text: case,

text: p

text: has

text: diﬀerent

text: dynamics

text: according

text: to

text: Deﬁnition

text: We

text: mention

text: below

text: some

text: open

text: question

text: related

text: to

text: the

text: previous

text: discussion.

text: Question

text: Is

text: there

text: an

text: example

text: of

text: a

text: pre-periodic

text: singular

text: end

text: point?

text: Question

text: Is

text: there

text: an

text: example

text: of

text: an

text: oscillating

text: singular

text: end

text: point

text: non

text: periodic

text: neither

text: pre-periodic?

text: Question

text: Is

text: there

text: an

text: example

text: of

text: an

text: escaping

text: hair

text: to

text: E

text: such

text: that

text: E

text: is

text: inﬁnite?

text: Example

text: of

text: escaping

text: hairs

text: with

text: wandering

text: singu-

text: lar

text: end

text: points

text: In

text: this

text: section

text: we

text: give

text: two

text: examples,

text: in

text: the

text: ﬁrst

text: one

text: we

text: show

text: that

text: for

text: each

text: of

text: the

text: ﬁnite

text: essential

text: singularities

text: of

text: the

text: function

text: g(z)

text: e

text: z,

text: has

text: inﬁnitely

text: many

text: Baker

text: sinz

text: domains

text: attached

text: to

text: it.

text: In

text: the

text: second

text: one,

text: for

text: each

text: ﬁnite

text: essential

text: singularity

text: of

text: the

text: function

text: f(z)

text: e

text: z

text: there

text: are

text: inﬁnitely

text: many

text: wandering

text: domains

text: attached

text: sinz

text: ##LTLine##

text: ##LTLine##


text: Figure 5: Escaping hairs with wandering singular end point on green and the singular end points on red in the dynamical plane. Figure 6: A detail of the dynamical plane of f and g showing the Fatou components Bj k close to zero. For g these components are Baker domains and zero is their absorbing point, for f they are wandering domains. 7.3 Escaping hairs with wandering singular end points It is clear that each Bj is contained in I (g), with absorbing point π2k, so they are k ± ⊂ g contained in I (g), while B∞ I∞(g). According to Deﬁnition 6.18, we have that π2k escaping points in all this Baker domains are not oscillating because they are escaping periodic of period 1. 28 ##LTFigure## ##LTFigure##

text: Figure

text: Escaping

label: caption

text: hairs

text: with

text: wandering

text: singular

text: end

text: point

text: on

text: green

text: and

text: the

text: singular

text: end

text: points

text: on

text: red

text: in

text: the

text: dynamical

text: plane.

text: Figure

text: A

text: detail

text: of

text: the

text: dynamical

text: plane

text: of

text: f

text: and

text: g

text: showing

text: the

text: Fatou

text: components

text: Bj

text: k

text: close

text: to

text: zero.

text: For

text: g

text: these

text: components

text: are

text: Baker

text: domains

text: and

text: zero

text: is

text: their

text: absorbing

text: point,

text: for

text: f

text: they

text: are

text: wandering

text: domains.

label: section

text: Escaping

text: hairs

text: with

text: wandering

text: singular

text: end

text: points

text: It

text: is

text: clear

text: that

text: each

text: Bj

text: is

text: contained

text: in

text: I

text: (g),

text: with

text: absorbing

text: point

text: so

text: they

text: are

text: k

text: g

text: contained

text: in

text: I

text: (g),

text: while

text: B∞

text: I∞(g).

text: According

text: to

text: Deﬁnition

text: we

text: have

text: that

text: escaping

text: points

text: in

text: all

text: this

text: Baker

text: domains

text: are

text: not

text: oscillating

text: because

text: they

text: are

text: escaping

text: periodic

text: of

text: period

text: ##LTFigure##

text: ##LTFigure##


text: In this paper the topics are divided as follows: in Section we deﬁne and study classes of functions, the set of singularities, exceptional values and examples of the deﬁnitions mentioned before. Section contains the iteration of the functions given in Section and a classiﬁcation of their compositions. In Section we review the deﬁnitions and K properties of the Fatou and Julia sets for functions in class and give some well known properties for those sets, including the classiﬁcation of the periodic components of the Fatou set. In Section 5 we show some results related to the dynamics of functions in K class . In Section 6 the escaping sets given by Eremenko in [38] are generalized in two ∈ diﬀerent kinds: I (f) which is the escaping set to a speciﬁc singularity e B(f) and the e general escaping set I (f) related to the whole B(f). We also classify the dynamics g in the general escaping set I (f) and extend the dynamics of f to the singularities g using curves contained in I (f). Section 7 contains examples of Baker and wandering g K domains for functions in class , we use such examples to construct escaping hairs with wandering singular end-points. K 2 -meromorphic functions (cid:98) (cid:98) C C In what follows we shall denote the complex plane by , the sphere by and the C∗ C \ { ∞} punctured plane by = 0, . We consider the following classes of functions. E { C → C | } = f : f is transcendental entire . P { C∗ → C∗ | } = f : f is transcendental holomorphic . P We shall distinguish the following two types of maps in class . P { ∈ P | } = f 0 is a pole and an omitted value 1 { ∈ N} = f(z) = eh(z) , h(z) entire non-constant, n . zn P { ∈ P | } = f 0 is an essential singularity 2 { ∈ Z} = f(z) = zne(g(z)+h(1/z)), g, h entire non-constant, n . (cid:98) C We recall that a function f is meromorphic in if f is analytic except at poles. M { C → C | } = f : f is transcendental meromorphic with one not omitted pole . A class of functions which contains all the classes mentioned above and their iterations is deﬁned a(cid:98)s follows. (cid:98) (cid:98) K { C \ → C | C \ } = f : B(f) f is not constant and meromorphic in B(f) , where the set B(f) is a compa(cid:98)ct countable set and it is the closure of the set of isolated essential singularities of f. We assume that B(f) has at least one element. In other ∈ K C \ words, f is analytic in B(f) except at poles. We are calling this functions K -meromorphic. 4 ##LTLine##

text: In

text: this

text: paper

text: the

text: topics

text: are

text: divided

text: as

text: follows:

text: in

text: Section

text: we

text: deﬁne

text: and

text: study

text: classes

text: of

text: functions,

text: the

text: set

text: of

text: singularities,

text: exceptional

text: values

text: and

text: examples

text: of

text: the

text: deﬁnitions

text: mentioned

text: before.

text: Section

text: contains

text: the

text: iteration

text: of

text: the

text: functions

text: given

text: in

text: Section

text: and

text: a

text: classiﬁcation

text: of

text: their

text: compositions.

text: In

text: Section

text: we

text: review

text: the

text: deﬁnitions

text: and

text: K

text: properties

text: of

text: the

text: Fatou

text: and

text: Julia

text: sets

text: for

text: functions

text: in

text: class

text: and

text: give

text: some

text: well

text: known

text: properties

text: for

text: those

text: sets,

text: including

text: the

text: classiﬁcation

text: of

text: the

text: periodic

text: components

text: of

text: the

text: Fatou

text: set.

text: In

text: Section

text: we

text: show

text: some

text: results

text: related

text: to

text: the

text: dynamics

text: of

text: functions

text: in

text: K

text: class

text: In

text: Section

text: the

text: escaping

text: sets

text: given

text: by

text: Eremenko

text: in

text: are

text: generalized

text: in

text: two

text: diﬀerent

text: kinds:

text: I

text: (f)

text: which

text: is

text: the

text: escaping

text: set

text: to

text: a

text: speciﬁc

text: singularity

text: e

text: B(f)

text: and

text: the

text: e

text: general

text: escaping

text: set

text: I

text: (f)

text: related

text: to

text: the

text: whole

text: B(f).

text: We

text: also

text: classify

text: the

text: dynamics

text: g

text: in

text: the

text: general

text: escaping

text: set

text: I

text: (f)

text: and

text: extend

text: the

text: dynamics

text: of

text: f

text: to

text: the

text: singularities

text: g

text: using

text: curves

text: contained

text: in

text: I

text: (f).

text: Section

text: contains

text: examples

text: of

text: Baker

text: and

text: wandering

text: g

text: K

text: domains

text: for

text: functions

text: in

text: class

text: we

text: use

text: such

text: examples

text: to

text: construct

text: escaping

text: hairs

text: with

text: wandering

text: singular

text: end-points.

text: K

text: -meromorphic

text: functions

text: C

text: C

text: In

text: what

text: follows

text: we

text: shall

text: denote

text: the

text: complex

text: plane

text: by

text: the

text: sphere

text: by

text: and

text: the

text: C∗

text: C

text: punctured

text: plane

text: by

text: We

text: consider

text: the

text: following

text: classes

text: of

text: functions.

text: E

text: C

text: C

text: f

text: f

text: is

text: transcendental

text: entire

text: P

text: C∗

text: C∗

text: f

text: f

text: is

text: transcendental

text: holomorphic

text: P

text: We

text: shall

text: distinguish

text: the

text: following

text: two

text: types

text: of

text: maps

text: in

text: class

text: P

text: P

text: f

text: is

text: a

text: pole

text: and

text: an

text: omitted

text: value

text: N}

text: f(z)

text: eh(z)

text: h(z)

text: entire

text: non-constant,

text: n

text: zn

text: P

text: P

text: f

text: is

text: an

text: essential

text: singularity

text: Z}

text: f(z)

text: g,

text: h

text: entire

text: non-constant,

text: n

text: C

text: We

text: recall

text: that

text: a

text: function

text: f

text: is

text: meromorphic

text: in

text: if

text: f

text: is

text: analytic

text: except

text: at

text: poles.

text: M

text: C

text: C

text: f

text: f

text: is

text: transcendental

text: meromorphic

text: with

text: one

text: not

text: omitted

text: pole

text: A

text: class

text: of

text: functions

text: which

text: contains

text: all

text: the

text: classes

text: mentioned

text: above

text: and

text: their

text: iterations

text: is

text: deﬁned

text: follows.

text: K

text: C

text: C

text: C

text: f

text: B(f)

text: f

text: is

text: not

text: constant

text: and

text: meromorphic

text: in

text: B(f)

text: where

text: the

text: set

text: B(f)

text: is

text: a

text: countable

text: set

text: and

text: it

text: is

text: the

text: closure

text: of

text: the

text: set

text: of

text: isolated

text: essential

text: singularities

text: of

text: f.

text: We

text: assume

text: that

text: B(f)

text: has

text: at

text: least

text: one

text: element.

text: In

text: other

text: K

text: C

text: words,

text: f

text: is

text: analytic

text: in

text: B(f)

text: except

text: at

text: poles.

text: We

text: are

text: calling

text: this

text: functions

text: K

text: -meromorphic.

text: ##LTLine##


text: ∈ K ∈ (b) If f and e B(f), then e is either an essential isolated singularity or an accumulation of those singularities. K (c) By the Picard Property, functions in class may have at most two Picard ex- ceptional values. (d) A point w is a Picard exceptional value of f if and only if w is a local Picard exceptional value of all singularities. Examples of functions having omitted values, Picard exceptional values and local Picard exceptional values are the following: ∞ (i) The exponential map ez has two omitted values, zero and which are also Picard exceptional values; ∞ (ii) The map ez/z has as a Picard exceptional value but it is no an omitted value, while zero is an omitted value; − ∞ (iii) The map e(z 1/z), has two omitted values, 0 and , such points are also essential singularities. { ∞} (iv) If f(z) = e1/z tan z, then B(f) = 0, and the set of local Picard exceptional { ∞} ∈ Z \ { } − ∞ values at zero is 0, . Since f(kπ) = 0 for all k 0 and f((2k 1)π/2) = ∈ Z ∞ for all k , therefore, 0 and are not Picard exceptional values of f. K Problem 2.4. Is there an example of a function in class , such that it has local Picard exceptional values at each essential singularity but not Picard exceptional values? Observe that such example cannot be meromorphic. ± Problem 2.5. Are the asymptotic values i Picard exceptional values of f in Example (iv)? If f is a function, the sequence formed by its iterates is denoted by f0 := Id, ◦ − ∈ N ◦ fn := f fn 1, n , where means composition and fn is the n-iterate of f. ∈ K Given f we deﬁne the forward orbit, the backward orbit and the grand orbit of { ∈ } − { a point z as O+(z) = w : fn(z) = w for n N , O (z) = w : fn(w) = z for some ∈ } ∪ − n N and O(z) = O+(z) O (z) respectively. Also, we deﬁne E(f) as the set of − Fatou exceptional values of f, this is, points whose inverse orbit O (z) is ﬁnite. For ∞ instance, take the map f(z) = ez, the Fatou exceptional values of f are zero and − (both are omitted values). Now observe the map h(z) = ez + 1/z, it has no omitted ∞ values, the Picard exceptional value is but it is not a Fatou exceptional value since − ∞ ∅ f n( ) is not ﬁnite, so E(h) = . As a result of the Iversen’s Theorem the corollary in [90] states the following. Corollary 2.6. Any Picard exceptional value of a transcendental meromorphic func- ∞ tion is also an asymptotic value. Hence, a transcendental entire function must have as its asymptotic value . K The above corollary can be extended to functions in class , in the sense that every local Picard exceptional value of f is an asymptotic value of f. It is clear that omitted 6

text: K

text: (b)

text: If

text: f

text: and

text: e

text: B(f),

text: then

text: e

text: is

text: either

text: an

text: essential

text: isolated

text: singularity

text: or

text: an

text: accumulation

text: of

text: those

text: singularities.

text: K

text: (c)

text: By

text: the

text: Picard

text: Property,

text: functions

text: in

text: class

text: may

text: have

text: at

text: most

text: two

text: Picard

text: ex-

text: ceptional

text: values.

text: (d)

text: A

text: point

text: w

text: is

text: a

text: Picard

text: exceptional

text: value

text: of

text: f

text: if

text: and

text: only

text: if

text: w

text: is

text: a

text: local

text: Picard

text: exceptional

text: value

text: of

text: all

text: singularities.

text: Examples

text: of

text: functions

text: having

text: omitted

text: values,

text: Picard

text: exceptional

text: values

text: and

text: local

text: Picard

text: exceptional

text: values

text: are

text: the

text: following:

text: (i)

text: The

text: exponential

text: map

text: ez

text: has

text: two

text: omitted

text: values,

text: zero

text: and

text: which

text: are

text: also

text: Picard

text: exceptional

text: values;

text: (ii)

text: The

text: map

text: ez/z

text: has

text: as

text: a

text: Picard

text: exceptional

text: value

text: but

text: it

text: is

text: no

text: an

text: omitted

text: value,

text: while

text: zero

text: is

text: an

text: omitted

text: value;

text: (iii)

text: The

text: map

text: e(z

text: has

text: two

text: omitted

text: values,

text: and

text: such

text: points

text: are

text: also

text: essential

text: singularities.

text: (iv)

text: If

text: f(z)

text: tan

text: z,

text: then

text: B(f)

text: and

text: the

text: set

text: of

text: local

text: Picard

text: exceptional

text: Z

text: values

text: at

text: zero

text: is

text: Since

text: f(kπ)

text: for

text: all

text: k

text: and

text: Z

text: for

text: all

text: k

text: therefore,

text: and

text: are

text: not

text: Picard

text: exceptional

text: values

text: of

text: f.

text: K

text: Problem

text: Is

text: there

text: an

text: example

text: of

text: a

text: function

text: in

text: class

text: such

text: that

text: it

text: has

text: local

text: Picard

text: exceptional

text: values

text: at

text: each

text: essential

text: singularity

text: but

text: not

text: Picard

text: exceptional

text: values?

text: Observe

text: that

text: such

text: example

text: cannot

text: be

text: meromorphic.

text: Problem

text: Are

text: the

text: asymptotic

text: values

text: i

text: Picard

text: exceptional

text: values

text: of

text: f

text: in

text: Example

text: (iv)?

text: If

text: f

text: is

text: a

text: function,

text: the

text: sequence

text: formed

text: by

text: its

text: iterates

text: is

text: denoted

text: by

text: Id,

text: N

text: fn

text: f

text: fn

text: n

text: where

text: means

text: composition

text: and

text: fn

text: is

text: the

text: n-iterate

text: of

text: f.

text: K

text: Given

text: f

text: we

text: deﬁne

text: the

text: forward

text: orbit,

text: the

text: backward

text: orbit

text: and

text: the

text: grand

text: orbit

text: of

text: a

text: point

text: z

text: as

text: O+(z)

text: w

text: fn(z)

text: w

text: for

text: n

text: N

text: O

text: (z)

text: w

text: fn(w)

text: z

text: for

text: some

text: n

text: N

text: and

text: O(z)

text: O+(z)

text: O

text: (z)

text: respectively.

text: Also,

text: we

text: deﬁne

text: E(f)

text: as

text: the

text: set

text: of

text: Fatou

text: exceptional

text: values

text: of

text: f,

text: this

text: is,

text: points

text: whose

text: inverse

text: orbit

text: O

text: (z)

text: is

text: ﬁnite.

text: For

text: instance,

text: take

text: the

text: map

text: f(z)

text: ez,

text: the

text: Fatou

text: exceptional

text: values

text: of

text: f

text: are

text: zero

text: and

text: (both

text: are

text: omitted

text: values).

text: Now

text: observe

text: the

text: map

text: h(z)

text: ez

text: it

text: has

text: no

text: omitted

text: values,

text: the

text: Picard

text: exceptional

text: value

text: is

text: but

text: it

text: is

text: not

text: a

text: Fatou

text: exceptional

text: value

text: since

text: f

text: n(

text: is

text: not

text: ﬁnite,

text: so

text: E(h)

text: As

text: a

text: result

text: of

text: the

text: Iversen’s

text: Theorem

text: the

text: corollary

text: in

text: states

text: the

text: following.

text: Corollary

text: Any

text: Picard

text: exceptional

text: value

text: of

text: a

text: transcendental

text: meromorphic

text: func-

text: tion

text: is

text: also

text: an

text: asymptotic

text: value.

text: Hence,

text: a

text: transcendental

text: entire

text: function

text: must

text: have

text: as

text: its

text: asymptotic

text: value

text: K

text: The

text: above

text: corollary

text: can

text: be

text: extended

text: to

text: functions

text: in

text: class

text: in

text: the

text: sense

text: that

text: every

text: local

text: Picard

text: exceptional

text: value

text: of

text: f

text: is

text: an

text: asymptotic

text: value

text: of

text: f.

text: It

text: is

text: clear

text: that

text: omitted


text: 13 (cid:104) (cid:105) Z− Then, we make use of the conservation law ∂ ηµην = 0 and expand in correlations (neglecting 2 order terms), t p p ﬁnding: (cid:0) (cid:1) (cid:88) (cid:0) (cid:1) G G G G ω µνα,β = iB (cid:15) µνθ,β + (cid:15) θνα,β + (cid:15) µθα,β (A3) ppn,n xαθ ppn,n xµθ ppn,n xνθ ppn,n (cid:104) (cid:105) (cid:104) (cid:105)G +(cid:15) (cid:0)V ηµηνηxηy Gθ,β + η(cid:1)xηy µνθ,β zαθ n,i p p i i C n,n i i ppn,n (cid:54) i(cid:88)=n,p (cid:16) (cid:17) (cid:104) (cid:105) − (cid:104) (cid:105)(cid:104) (cid:105) +(cid:15) V ηµηνηxηy ηµην ηxηy Gθ,β zαθ n,p p p p p p p p p n,n +(cid:15) (cid:88) V (cid:16)(cid:104)ηxηy(cid:105)Gθνα,β + (cid:104)ηθην(cid:105)Gxyα,β(cid:17) zµθ p,i i i ppn,n p p iin,n (cid:54) i=p,n (cid:104) (cid:105)G (cid:104) (cid:105)G +(cid:15) (cid:0) V ηxηy µθα,β +(cid:1) (cid:0)ηµηθ xyα,β (cid:1) zνθ p,i i i ppn,n p p iin,n (cid:54) i=p,n (cid:104) (cid:105) (cid:104) (cid:105) − (cid:104) (cid:105) +V (cid:15) ηθην + (cid:15) ηµηθ Gxyα,β ηxηy Gα,β p,n zµθ p p zνθ p p nnn,n n n n,n (cid:104) (cid:105) (cid:104) (cid:105) Importantly, if we make use of the properties of the transverse Ising model Sy = 0 and SzSy = 0, we can simplify i i j the diﬀerent concrete cases. Also, the Green’s function Gxyα,β appear in the equation of motion, but fortunately it nnn,n can be calculated exactly: 1 (cid:104) (cid:105) − (cid:104) (cid:105) (cid:104) (cid:105) Gxyz,β = (δ ηyηz δ ηxηz + δ ηxηy ) (A4) nnn,n 2πω x,β n n y,β n n z,β n n Appendix B: Classical Ising model The full equation of motion in momentum space is given by: (cid:88) N 1 G ωGα,β = δ (k) δ + iB(cid:15) Gµ,β + (cid:15) iM V Gµ,β + (cid:15) V xyµ−,β (B1) k 2π α,β xαµ k zαµ z 0 k zαµ N q q,k q q Neglecting the tran(cid:18)sverse term and wr(cid:19)it(cid:18)ing the (cid:19)diﬀerent com(cid:18)ponen(cid:19)ts in matrix f(cid:32)orm we ﬁnd(cid:33): (cid:88) − G ω iM V Gx,β δ δ 1 xyy−,β z 0 k = N k,0 x,β + V −Gq,k q (B2) iMzV0 ω Gy,β 2π δy,β N q xyx−,β k q q,k q Solving the equation of motion for the co(cid:0)rrelated(cid:1)part and assuming that the magnetization is homogeneous we (cid:16) (cid:17) ﬁnd: − − (cid:104) (cid:105) Gkx,ykx(cid:48),β = Vk(cid:48) Nδk,−k(cid:48) (cid:0)41 ω2M−z2V(cid:1) 2Mη2kxηkyηkx(cid:48)ηky(cid:48) C (cid:16)iV0MzGx0,β − ωGy0,β(cid:17) − 0− (cid:104)z (cid:105) Gkx,yky(cid:48),β = Vk(cid:48) Nδk,−k(cid:48) 41 ω2M−z2V 2Mη2kxηkyηkx(cid:48)ηky(cid:48) C iV0MzGy0,β + ωGx0,β 0 z where Gα,β = Gα,βδ (k) /N. Then we can rewrite the initial equation of motion as: k 0 (cid:16) (cid:17) − δˆ · (cid:18) (cid:19) (cid:18) (cid:19) (cid:18)ω Hˆ0 Gˆ = 2π +(cid:19)Σˆ (ω) Gˆ(cid:16) (cid:17) (B3) − Gx,β δ 0 iM V − 1 · being Gˆ = 0 , δˆ = x,β , Hˆ = − z 0 , Gˆ(0) = ω Hˆ δˆ and Gy0,β δy,β 0 iMzV0 0 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Z−

text: Then,

text: we

text: make

text: use

text: of

text: the

text: conservation

text: law

text: ηµην

text: and

text: expand

text: in

text: correlations

text: (neglecting

text: order

text: terms),

text: t

text: p

text: p

text: ﬁnding:

text: G

text: G

text: G

text: G

text: ω

text: µνα,β

text: iB

text: µνθ,β

text: θνα,β

text: µθα,β

text: ppn,n

text: xαθ

text: ppn,n

text: xµθ

text: ppn,n

text: xνθ

text: ppn,n

text: ηµηνηxηy

text: Gθ,β

text: µνθ,β

text: zαθ

text: n,i

text: p

text: p

text: i

text: i

text: C

text: n,n

text: i

text: i

text: ppn,n

font: default

text: V

text: ηµηνηxηy

text: ηµην

text: ηxηy

text: Gθ,β

text: zαθ

text: n,p

text: p

text: p

text: p

text: p

text: p

text: p

text: p

text: p

text: n,n

text: V

text: zµθ

text: p,i

text: i

text: i

text: ppn,n

text: p

text: p

text: iin,n

text: i=p,n

text: V

text: ηxηy

text: µθα,β

text: xyα,β

text: zνθ

text: p,i

text: i

text: i

text: ppn,n

text: p

text: p

text: iin,n

text: i=p,n

text: +V

text: ηθην

text: ηµηθ

text: Gxyα,β

text: ηxηy

text: Gα,β

text: p,n

text: zµθ

text: p

text: p

text: zνθ

text: p

text: p

text: nnn,n

text: n

text: n

text: n,n

text: Importantly,

text: if

text: we

text: make

text: use

text: of

text: the

text: properties

text: of

text: the

text: transverse

text: Ising

text: model

text: Sy

text: and

text: SzSy

text: we

text: can

text: simplify

text: i

text: i

text: j

text: the

text: diﬀerent

text: concrete

text: cases.

text: Also,

text: the

text: Green’s

text: function

text: Gxyα,β

text: appear

text: in

text: the

text: equation

text: of

text: motion,

text: but

text: fortunately

text: it

text: nnn,n

text: can

text: be

text: calculated

text: exactly:

text: Gxyz,β

text: (δ

text: ηyηz

text: δ

text: ηxηz

text: δ

text: ηxηy

text: nnn,n

text: 2πω

text: x,β

text: n

text: n

text: y,β

text: n

text: n

text: z,β

text: n

text: n

text: Appendix

text: B:

text: Classical

text: Ising

text: model

text: The

text: full

text: equation

text: of

text: motion

text: in

text: momentum

text: space

text: is

text: given

text: by:

text: N

text: G

text: ωGα,β

text: δ

text: (k)

text: δ

text: Gµ,β

text: iM

text: V

text: Gµ,β

text: V

text: xyµ−,β

text: k

text: 2π

text: α,β

text: xαµ

text: k

text: zαµ

text: z

text: k

text: zαµ

text: N

text: q

text: q,k

text: q

text: q

text: Neglecting

text: the

text: term

text: and

text: the

text: in

text: matrix

text: we

text: G

text: ω

text: iM

text: V

text: Gx,β

text: δ

text: δ

text: xyy−,β

text: z

text: k

text: N

text: x,β

text: V

text: −Gq,k

text: q

text: ω

text: Gy,β

text: 2π

text: δy,β

text: N

text: q

text: xyx−,β

text: k

text: q

text: q,k

text: q

text: Solving

text: the

text: equation

text: of

text: motion

text: for

text: the

text: and

text: assuming

text: that

text: the

text: magnetization

text: is

text: homogeneous

text: we

text: ﬁnd:

text: C

text: 0−

text: C

text: z

text: where

text: Gα,β

text: Gα,βδ

text: (k)

text: /N.

text: Then

text: we

text: can

text: rewrite

text: the

text: initial

text: equation

text: of

text: motion

text: as:

text: k

text: δˆ

text: Gˆ

text: 2π

text: (ω)

text: Gx,β

text: δ

text: iM

text: V

text: being

text: Gˆ

text: δˆ

text: x,β

text: Hˆ

text: z

text: ω

text: Hˆ

text: δˆ

text: and

text: δy,β

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 11 TABLE I: Physical parameters and initial conditions for selected case studies System m m m a (a.u.) A(a.u.) e E z ω ω 1 2 3 3 Hot Jupiters M M(cid:12) 40 M 6 100 0.001 0.6 65 45 0 J J Coplanar Flips M M(cid:12) 0.03 M(cid:12) 4 50 0.8 0.6 5 0 0 J Asteroid-Jupiter 0 M(cid:12) M 2 5 0.2 0.05 65 0 0 J Triple star 0.25 M(cid:12) M(cid:12) 0.6 M(cid:12) 60 800 0.01 0.6 98 0 0 CH Cygni 0.5 M(cid:12) 3.51 M(cid:12) 0.909 M(cid:12) 0.05 0.21 0.32 0.6 72 145 0 B. Orbital ﬂips from nearly coplanar orbits 180 135 Li et al. [8] discovered the possibility of generating or- bital ﬂips and large eccentricities from initially nearly 90 coplanar orbits using the octupole-order equations. The 45 inner system was again a Jupiter-Sun binary with a = 4 a.u., perturbed by a brown dwarf, with m = 0.03 M(cid:12) 3 and A = 50 a.u. The parameters then have the values 10-2 × − α = 0.030 (cid:15) = 0.08 , η = 9.53 10 4 , 10-4 × − octupole ∆ = 0.998 , δ = 1.63 10 9 , (3.5) hexadecapole 10-6 hexadecapole & GR 0 0.5 1 1.5 2 time (Myr) and the initial conditions are FIG. 3: Orbital ﬂips and eccentricity excursions in a nearly e = 0.8 , E = 0.6 , z = 5o , ω = 0o , ω = 0o . (3.6) 3 co-planar Jupiter system. Black: octupole order. Blue: hex- adecapole order. Red: hexadecapole order with GR. Param- × eters and initial orbit elements are listed in Table I. (Color We evolve the equations for 2.5 105 inner orbits × ﬁgures in online version.) (2 106 years), for three cases: octupole and hexade- capole orders without GR precessions, and hexadecapole order with GR precessions. The results are shown in Fig. 3. At octupole order without GR (upper panel, plotted parameters and initial conditions ﬁxed, we obtain the in black), the system oscillates about small values of z curves shown in Fig. 4. Here the time scales as τ(a/4)3, for a while, then migrates quickly to a retrograde orbit, where a is in astronomical units; this timescale is chosen oscillates about the new values for a while, then migrates so that similar numbers of Kozai cycles can appear on back. During the transition the eccentricity reaches ex- one plot. For a = 5 (blue), the pattern of ﬂips is very treme values close to unity (lower panel). Including the similar to that obtained without GR, shown in blue in hexadecapole terms shortens the timescale slightly (plot- Fig. 3. For a = 4 (red), the curve is the same as that ted in blue), but otherwise preserves the basic behavior. shown in red in Fig. 3. For a = 3 (green) the migration These curves are in excellent agreement with results ob- to large inclinations is suppressed by the more rapid GR tained by Li [20] and by Hamers [21] using both N-body precessions, although migrations to large eccentricities codes and orbit element codes to the same multipolar still occur. Finally, for a = 2, (black) the GR precessions order. However, including the GR precessions with the permit only small Kozai-like oscillations about the initial hexadecapole terms causes the ﬁrst ﬂip to abort (plotted values of z and e. in red); subsequent ﬂips are then out of phase with those For the nominal value a = 4 a.u., we also show the where GR is not included. It is evident that the transi- sensitivity of orbital ﬂips to the pericenter angles. Fig- tion from prograde to retrograde orbits is very sensitive ure 5 shows evolutions for four initial pericenter angles to the phases of the two pericenter angles, ω and ω as of body 3: 0o (red, same as in Fig. 3), 45o (blue), 90o 3 the inclination angle z approaches 90o. The cumulative (green) and 180o (black). Notice that the initial values precessions in these angles induced by general relativity ω = 0o, ω = 0o correspond to orbits with the initial 3 can turn a transition to retrograde into a bounce back to pericenters pointing in opposite directions (Fig. 1), while prograde, and vice versa. the values ω = 0o, ω = 180o correspond to initial orbits 3 We now vary the semimajor axis a of the inner orbit with the pericenters pointing in the same direction. This in order to assess the eﬀects of GR. Holding the other dependence is in agreement with results from N-body )ged( z e - 1 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: I:

text: Physical

text: parameters

text: and

text: initial

text: conditions

text: for

text: selected

text: case

text: studies

text: System

label: table

text: m

text: m

text: m

text: a

text: (a.u.)

text: A(a.u.)

text: e

text: E

text: z

text: ω

text: ω

text: Hot

text: Jupiters

text: M

text: M

text: J

text: J

text: Coplanar

text: Flips

text: M

text: J

text: Asteroid-Jupiter

text: M

text: J

text: Triple

text: star

text: CH

text: Cygni

text: B.

text: Orbital

text: ﬂips

text: from

text: nearly

text: coplanar

text: orbits

text: Li

text: et

text: al.

text: discovered

text: the

text: possibility

text: of

text: generating

text: or-

text: bital

text: ﬂips

text: and

text: large

text: eccentricities

text: from

text: initially

text: nearly

text: coplanar

text: orbits

text: using

text: the

text: octupole-order

text: equations.

text: The

text: inner

text: system

text: was

text: again

text: a

text: Jupiter-Sun

text: binary

text: with

text: a

text: a.u.,

text: perturbed

text: by

text: a

text: brown

text: dwarf,

text: with

text: m

text: and

text: A

text: a.u.

text: The

text: parameters

text: then

text: have

text: the

text: values

text: α

text: η

text: octupole

text: δ

text: hexadecapole

text: hexadecapole

text: GR

text: time

text: (Myr)

text: and

text: the

text: initial

text: conditions

text: are

text: FIG.

text: Orbital

text: ﬂips

text: and

text: eccentricity

text: excursions

text: in

text: a

text: nearly

text: e

text: E

text: z

text: ω

text: ω

text: co-planar

text: Jupiter

text: system.

text: Black:

text: octupole

text: order.

text: Blue:

text: hex-

text: adecapole

text: order.

text: Red:

text: hexadecapole

text: order

text: with

text: GR.

text: Param-

text: eters

text: and

text: initial

text: orbit

text: elements

text: are

text: listed

text: in

text: Table

text: I.

text: (Color

text: We

text: evolve

text: the

text: equations

text: for

text: inner

text: orbits

text: ﬁgures

text: in

text: online

text: version.)

text: years),

text: for

text: three

text: cases:

text: octupole

text: and

text: hexade-

text: capole

text: orders

text: without

text: GR

text: precessions,

text: and

text: hexadecapole

text: order

text: with

text: GR

text: precessions.

text: The

text: results

text: are

text: shown

text: in

text: Fig.

text: At

text: octupole

text: order

text: without

text: GR

text: (upper

text: panel,

text: plotted

text: parameters

text: and

text: initial

text: conditions

text: ﬁxed,

text: we

text: obtain

text: the

text: in

text: black),

text: the

text: system

text: oscillates

text: about

text: small

text: values

text: of

text: z

text: curves

text: shown

text: in

text: Fig.

text: Here

text: the

text: time

text: scales

text: as

text: for

text: a

text: while,

text: then

text: migrates

text: quickly

text: to

text: a

text: retrograde

text: orbit,

text: where

text: a

text: is

text: in

text: astronomical

text: units;

text: this

text: timescale

text: is

text: chosen

text: oscillates

text: about

text: the

text: new

text: values

text: for

text: a

text: while,

text: then

text: migrates

text: so

text: that

text: similar

text: numbers

text: of

text: Kozai

text: cycles

text: can

text: appear

text: on

text: back.

text: During

text: the

text: transition

text: the

text: eccentricity

text: reaches

text: ex-

text: one

text: plot.

text: For

text: a

text: (blue),

text: the

text: pattern

text: of

text: ﬂips

text: is

text: very

text: treme

text: values

text: close

text: to

text: unity

text: (lower

text: panel).

text: Including

text: the

text: similar

text: to

text: that

text: obtained

text: without

text: GR,

text: shown

text: in

text: blue

text: in

text: hexadecapole

text: terms

text: shortens

text: the

text: timescale

text: slightly

text: (plot-

text: Fig.

text: For

text: a

text: (red),

text: the

text: curve

text: is

text: the

text: same

text: as

text: that

text: ted

text: in

text: blue),

text: but

text: otherwise

text: preserves

text: the

text: basic

text: behavior.

text: shown

text: in

text: red

text: in

text: Fig.

text: For

text: a

text: (green)

text: the

text: migration

text: These

text: curves

text: are

text: in

text: excellent

text: agreement

text: with

text: results

text: ob-

text: to

text: large

text: inclinations

text: is

text: suppressed

text: by

text: the

text: more

text: rapid

text: GR

text: tained

text: by

text: Li

text: and

text: by

text: Hamers

text: using

text: both

text: N-body

text: precessions,

text: although

text: migrations

text: to

text: large

text: eccentricities

text: codes

text: and

text: orbit

text: element

text: codes

text: to

text: the

text: same

text: multipolar

text: still

text: occur.

text: Finally,

text: for

text: a

text: (black)

text: the

text: GR

text: precessions

text: order.

text: However,

text: including

text: the

text: GR

text: precessions

text: with

text: the

text: permit

text: only

text: small

text: Kozai-like

text: oscillations

text: about

text: the

text: initial

text: hexadecapole

text: terms

text: causes

text: the

text: ﬁrst

text: ﬂip

text: to

text: abort

text: (plotted

text: values

text: of

text: z

text: and

text: e.

text: in

text: red);

text: subsequent

text: ﬂips

text: are

text: then

text: out

text: of

text: phase

text: with

text: those

text: For

text: the

text: nominal

text: value

text: a

text: a.u.,

text: we

text: also

text: show

text: the

text: where

text: GR

text: is

text: not

text: included.

text: It

text: is

text: evident

text: that

text: the

text: transi-

text: sensitivity

text: of

text: orbital

text: ﬂips

text: to

text: the

text: pericenter

text: angles.

text: Fig-

text: tion

text: from

text: prograde

text: to

text: retrograde

text: orbits

text: is

text: very

text: sensitive

text: ure

text: shows

text: evolutions

text: for

text: four

text: initial

text: pericenter

text: angles

text: to

text: the

text: phases

text: of

text: the

text: two

text: pericenter

text: angles,

text: ω

text: and

text: ω

text: as

text: of

text: body

text: (red,

text: same

text: as

text: in

text: Fig.

text: (blue),

text: the

text: inclination

text: angle

text: z

text: approaches

text: The

text: cumulative

text: (green)

text: and

text: (black).

text: Notice

text: that

text: the

text: initial

text: values

text: precessions

text: in

text: these

text: angles

text: induced

text: by

text: general

text: relativity

text: ω

text: ω

text: correspond

text: to

text: orbits

text: with

text: the

text: initial

text: can

text: turn

text: a

text: transition

text: to

text: retrograde

text: into

text: a

text: bounce

text: back

text: to

text: pericenters

text: pointing

text: in

text: opposite

text: directions

text: (Fig.

text: while

text: prograde,

text: and

text: vice

text: versa.

text: the

text: values

text: ω

text: ω

text: correspond

text: to

text: initial

text: orbits

text: We

text: now

text: vary

text: the

text: semimajor

text: axis

text: a

text: of

text: the

text: inner

text: orbit

text: with

text: the

text: pericenters

text: pointing

text: in

text: the

text: same

text: direction.

text: This

text: in

text: order

text: to

text: assess

text: the

text: eﬀects

text: of

text: GR.

text: Holding

text: the

text: other

text: dependence

text: is

text: in

text: agreement

text: with

text: results

text: from

text: N-body

text: )ged(

text: z

text: e

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 180 135 135 90 90 45 45 10-2 10-2 10-4 a = 2 a = 3 octupole a = 4 hexadecapole a = 5 10-6 0.5 tim1e.0 (arbitrary u1n.5its) 2.0 2.5 10-40 0.5 1 1.5 2 2.5 time (Myr) FIG. 4: Eﬀects of GR on orbital ﬂips and eccentricity excur- FIG. 6: Evolution of inclination and eccentricity in an sions in a nearly co-planar Jupiter system. Blue: a = 5 a.u. asteroid-Jupiter system. Blue: octupole order. Red: hex- Red: a = 4 a.u. (same as in Fig. 3). Green: a = 3 a.u. Black: adecapole order. GR precessions are included. (Color ﬁgures a = 2 a.u. Other parameters and initial orbit elements are in online version.) listed in Table I. (Color ﬁgures in online version.) 180 The initial conditions are ω = 00 3 ω = 45ο 135 ω33 = 90ο e = 0.2 , E = 0.05 , z = 65o , ω = 0o , ω = 0o . (3.8) ω = 180ο 3 3 90 We evolve the planetary equations for 106 orbital peri- ods, corresponding to about 2.8 million years, with and without hexadecapole terms. We include the GR preces- 45 sions, but they turn out to have negligible eﬀect in this example. Figure 6 shows the resulting evolutions of z and e. Including the hexadecapole terms stretches the timescale somewhat, in agreement with the full N-body 0 0.5 1 1.5 2 time (Myr) numerical evolutions carried out by NFLRT (see Fig. 8 of NFLRT). As in the previous example, the initial choice ω = 180o leads to no orbital ﬂips. FIG. 5: Eﬀects of initial pericenter angle on orbital ﬂips in a 3 D. A triple-star hierarchical system nearly co-planar Jupiter system for a = 4 a.u. Red: ω = 0o 3 (same as in Fig. 3). Blue: ω = 45o. Green: ω = 90o. Black: 3 3 ω = 180o. Other parameters and initial orbit elements are Analyzing a set of hierarchical triple-star systems stud- 3 listed in Table I. (Color ﬁgures in online version.) ied by Fabrycky and Tremaine [22], NFLRT again found orbital ﬂip behavior (Fig. 9 of [7]). The system studied consists of an inner binary with m = 0.25M(cid:12), m = 1 2 M(cid:12), a = 60 a.u., and an outer star, with m = 0.6M(cid:12), integrations by Li [20]. 3 a = 800 a.u. In this example, the parameters are C. An asteroid Jupiter system NFLRT showed that octupole perturbations could in- α = 0.48 (cid:15) = 0.075 , η = 0.16 , duce orbital ﬂips in a Sun-asteroid-Jupiter system. In × − ∆ = 0.6 , δ = 2.05 10 10 . (3.9) this case, a = 2 a.u. and A = 5 a.u., and the parameters are The initial conditions are × − α = 9.55 10 4 (cid:15) = 0.4 , η = 0 , × − ∆ = 1 , δ = 4.92 10 9 . (3.7) e = 0.01 , E = 0.6 , z = 98o , ω = 0o , ω = 0o . (3.10) 3 )ged( z e - 1 )ged( z )ged( z e - 1 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: a

text: a

text: octupole

text: a

text: hexadecapole

text: a

text: (arbitrary

text: time

text: (Myr)

text: FIG.

text: Eﬀects

text: of

text: GR

text: on

text: orbital

text: ﬂips

text: and

text: eccentricity

text: excur-

text: FIG.

text: Evolution

text: of

text: inclination

text: and

text: eccentricity

text: in

text: an

text: sions

text: in

text: a

text: nearly

text: co-planar

text: Jupiter

text: system.

text: Blue:

text: a

text: a.u.

text: asteroid-Jupiter

text: system.

text: Blue:

text: octupole

text: order.

text: Red:

text: hex-

text: Red:

text: a

text: a.u.

text: (same

text: as

text: in

text: Fig.

text: Green:

text: a

text: a.u.

text: Black:

text: adecapole

text: order.

text: GR

text: precessions

text: are

text: included.

text: (Color

text: ﬁgures

text: a

text: a.u.

text: Other

text: parameters

text: and

text: initial

text: orbit

text: elements

text: are

text: in

text: online

text: version.)

text: listed

text: in

text: Table

text: I.

text: (Color

text: ﬁgures

text: in

text: online

text: version.)

text: The

text: initial

text: conditions

text: are

text: ω

text: ω

text: e

text: E

text: z

text: ω

text: ω

text: ω

text: We

text: evolve

text: the

text: planetary

text: equations

text: for

text: orbital

text: peri-

text: ods,

text: corresponding

text: to

text: about

text: million

text: years,

text: with

text: and

text: without

text: hexadecapole

text: terms.

text: We

text: include

text: the

text: GR

text: preces-

text: sions,

text: but

text: they

text: turn

text: out

text: to

text: have

text: negligible

text: eﬀect

text: in

text: this

text: example.

text: Figure

text: shows

text: the

text: resulting

text: evolutions

text: of

text: z

text: and

text: e.

text: Including

text: the

text: hexadecapole

text: terms

text: stretches

text: the

text: timescale

text: somewhat,

text: in

text: agreement

text: with

text: the

text: full

text: N-body

text: time

text: (Myr)

text: numerical

text: evolutions

text: carried

text: out

text: by

text: NFLRT

text: (see

text: Fig.

text: of

text: NFLRT).

text: As

text: in

text: the

text: previous

text: example,

text: the

text: initial

text: choice

text: ω

text: leads

text: to

text: no

text: orbital

text: ﬂips.

text: FIG.

text: Eﬀects

text: of

text: initial

text: pericenter

text: angle

text: on

text: orbital

text: ﬂips

text: in

text: a

text: D.

text: A

text: triple-star

text: hierarchical

text: system

text: nearly

text: co-planar

text: Jupiter

text: system

text: for

text: a

text: a.u.

text: Red:

text: ω

text: (same

text: as

text: in

text: Fig.

text: Blue:

text: ω

text: Green:

text: ω

text: Black:

text: ω

text: Other

text: parameters

text: and

text: initial

text: orbit

text: elements

text: are

text: Analyzing

text: a

text: set

text: of

text: hierarchical

text: triple-star

text: systems

text: stud-

text: listed

text: in

text: Table

text: I.

text: (Color

text: ﬁgures

text: in

text: online

text: version.)

text: ied

text: by

text: Fabrycky

text: and

text: Tremaine

text: NFLRT

text: again

text: found

text: orbital

text: ﬂip

text: behavior

text: (Fig.

text: of

text: The

text: system

text: studied

text: consists

text: of

text: an

text: inner

text: binary

text: with

text: m

text: m

text: a

text: a.u.,

text: and

text: an

text: outer

text: star,

text: with

text: m

text: integrations

text: by

text: Li

text: a

text: a.u.

text: In

text: this

text: example,

text: the

text: parameters

text: are

text: C.

text: An

text: asteroid

text: Jupiter

text: system

text: NFLRT

text: showed

text: that

text: octupole

text: perturbations

text: could

text: in-

text: α

text: η

text: duce

text: orbital

text: ﬂips

text: in

text: a

text: Sun-asteroid-Jupiter

text: system.

text: In

text: δ

text: this

text: case,

text: a

text: a.u.

text: and

text: A

text: a.u.,

text: and

text: the

text: parameters

text: are

text: The

text: initial

text: conditions

text: are

text: α

text: η

text: δ

text: e

text: E

text: z

text: ω

text: ω

text: )ged(

text: z

text: e

text: )ged(

text: z

text: )ged(

text: z

text: e

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 180 135 135 90 90 45 45 10-2 10-2 10-4 10-4 ω3 == 09o0 oor 180o 10-6 qhueaxaddruepcoalpeole ω3 10-6 100 200 300 400 500 10-80 100 200 300 400 500 600 700 number of orbits number of orbits FIG. 9: Evolution of z and e for an equal-mass inner binary. FIG. 10: Evolution of z and e for an equal-mass inner bi- Red: ω3 = 0o or 180o. Blue: ω3 = 90o. GR precessions are nary in a nearly coplanar initial orbit. Blue: quadrupole or- included. (Color ﬁgures in online version.) der. Red: hexadecapole order. GR precessions are included. (Color ﬁgures in online version.) ω = 0o or 180o (red) , 90o (blue) . (4.2) 3 parameters are A speciﬁc system with these parameters consists of two × − 1.4 M(cid:12) neutron stars orbiting a 30 M(cid:12) star or black hole, α = 17.857 , (cid:15) = 0.0875 , δ = 3.95 10 9 . (4.3) with a = 7 a.u. and A = 55 a.u. Scaling all masses and semimajor axes by a common factor ζ yields identical evolutions, since the three parameters of Eq. (4.1) are A sample system is again two 1.4 M(cid:12) neutron stars unchanged. Only the timescale set by the inner orbital with a = 7 a.u., but now orbiting a 50 M(cid:12) star or black period changes, scaling by ζ. Evolving the system for 500 hole at A = 80 a.u. The initial conditions are orbits of the inner binary, we ﬁnd that the evolution for ω = 0o (initial pericenters in opposite directions along 3 e = 0.99 , E = 0.6 , z = 5o , ω = 45o , ω = 0o . (4.4) the line of nodes) is identical to that for ω = 180o (initial 3 3 pericenters in the same direction), resulting in an orbital ﬂip (red curves in Fig. 9), while the evolution for ω = 3 The quadrupole-order evolution, shown in blue, dis- 90o does not show ﬂips. This is in contrast to the cases plays the standard Kozai-Lidov cycles, whereas the where octupole terms dominate, where ω = 0o leads 3 hexadecapole-order evolution shows orbital ﬂips and ex- to ﬂips while ω = 180o does not. This makes sense 3 cursions to extreme eccentricities, well beyond (in the because, as can be seen from Eqs. (2.27), the octupole − → sense of log(1 e)) the initial relatively large initial value terms change sign under the transformation ω ω + 3 3 of e = 0.99. Here again, GR precessions play a negligible π, whereas the hexadecapole terms in Eqs. (2.28) are role, suppressing the ﬂips only when the GR parameter invariant under that transformation. On the other hand, δ is dialed up by a factor of about 80. many pieces of the hexadecapole terms change sign under → As a ﬁnal example, we display in Fig. 11 the eﬀect of the transformation ω ω +π/2, and as a consequence, 3 3 slightly unequal masses on the generation of orbital ﬂips the initial angle ω = 90o yields no ﬂips. 3 via octupole-order terms. We again consider an inner In the foregoing example, the evolutions are the same binary of total mass 2.8 M(cid:12), with a = 4 a.u., orbiting a whether the GR precessions are included or not. We star or black hole of mass 50 M(cid:12) at A = 50 a.u. The can investigate when GR eﬀects become important by initial conditions are “dialing up” the GR parameter δ while holding α and (cid:15) ﬁxed. This is equivalent either to reducing a and A by the same factor, holding the masses ﬁxed, or to increasing e = 0.8 , E = 0.6 , z = 75o , ω = 0o , ω = 0o . (4.5) 3 all the masses by the same factor, holding a and A ﬁxed. We ﬁnd that orbital ﬂips are preserved until δ is about 820 times larger than the value shown in Eq. (4.1). The equal-mass case shows no orbital ﬂips in this case Another example generates orbital ﬂips from nearly (blue curves in Fig. 11), basically because (cid:15) = 0.08 is coplanar orbits, an analogue of the case discussed in Sec. smaller than in the previous cases, and the hexadecopole . The results are shown in Fig. 10. In this case the terms alone are not large enough to do the job. As we )ged( z e - 1 )ged( z e - 1 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ω3

text: oor

text: qhueaxaddruepcoalpeole

text: ω3

text: number

text: of

text: orbits

text: number

text: of

text: orbits

text: FIG.

text: Evolution

text: of

text: z

text: and

text: e

text: for

text: an

text: equal-mass

text: inner

text: binary.

text: FIG.

text: Evolution

text: of

text: z

text: and

text: e

text: for

text: an

text: equal-mass

text: inner

text: bi-

text: Red:

text: ω3

text: or

text: Blue:

text: ω3

text: GR

text: precessions

text: are

text: nary

text: in

text: a

text: nearly

text: coplanar

text: initial

text: orbit.

text: Blue:

text: quadrupole

text: or-

text: included.

text: (Color

text: ﬁgures

text: in

text: online

text: version.)

text: der.

text: Red:

text: hexadecapole

text: order.

text: GR

text: precessions

text: are

text: included.

text: (Color

text: ﬁgures

text: in

text: online

text: version.)

text: ω

text: or

text: (red)

text: (blue)

text: parameters

text: are

text: A

text: speciﬁc

text: system

text: with

text: these

text: parameters

text: consists

text: of

text: two

text: neutron

text: stars

text: orbiting

text: a

text: star

text: or

text: black

text: hole,

text: α

text: δ

text: with

text: a

text: a.u.

text: and

text: A

text: a.u.

text: Scaling

text: all

text: masses

text: and

text: semimajor

text: axes

text: by

text: a

text: common

text: factor

text: ζ

text: yields

text: identical

text: evolutions,

text: since

text: the

text: three

text: parameters

text: of

text: Eq.

text: are

text: A

text: sample

text: system

text: is

text: again

text: two

text: neutron

text: stars

text: unchanged.

text: Only

text: the

text: timescale

text: set

text: by

text: the

text: inner

text: orbital

text: with

text: a

text: a.u.,

text: but

text: now

text: orbiting

text: a

text: star

text: or

text: black

text: period

text: changes,

text: scaling

text: by

text: ζ.

text: Evolving

text: the

text: system

text: for

text: hole

text: at

text: A

text: a.u.

text: The

text: initial

text: conditions

text: are

text: orbits

text: of

text: the

text: inner

text: binary,

text: we

text: ﬁnd

text: that

text: the

text: evolution

text: for

text: ω

text: (initial

text: pericenters

text: in

text: opposite

text: directions

text: along

text: e

text: E

text: z

text: ω

text: ω

text: the

text: line

text: of

text: nodes)

text: is

text: identical

text: to

text: that

text: for

text: ω

text: (initial

text: pericenters

text: in

text: the

text: same

text: direction),

text: resulting

text: in

text: an

text: orbital

text: ﬂip

text: (red

text: curves

text: in

text: Fig.

text: while

text: the

text: evolution

text: for

text: ω

text: The

text: quadrupole-order

text: evolution,

text: shown

text: in

text: blue,

text: dis-

text: does

text: not

text: show

text: ﬂips.

text: This

text: is

text: in

text: contrast

text: to

text: the

text: cases

text: plays

text: the

text: standard

text: Kozai-Lidov

text: cycles,

text: whereas

text: the

text: where

text: octupole

text: terms

text: dominate,

text: where

text: ω

text: leads

text: hexadecapole-order

text: evolution

text: shows

text: orbital

text: ﬂips

text: and

text: ex-

text: to

text: ﬂips

text: while

text: ω

text: does

text: not.

text: This

text: makes

text: sense

text: cursions

text: to

text: extreme

text: eccentricities,

text: well

text: beyond

text: (in

text: the

text: because,

text: as

text: can

text: be

text: seen

text: from

text: Eqs.

text: the

text: octupole

text: sense

text: of

text: e))

text: the

text: initial

text: relatively

text: large

text: initial

text: value

text: terms

text: change

text: sign

text: under

text: the

text: transformation

text: ω

text: ω

text: of

text: e

text: Here

text: again,

text: GR

text: precessions

text: play

text: a

text: negligible

text: π,

text: whereas

text: the

text: hexadecapole

text: terms

text: in

text: Eqs.

text: are

text: role,

text: suppressing

text: the

text: ﬂips

text: only

text: when

text: the

text: GR

text: parameter

text: invariant

text: under

text: that

text: transformation.

text: On

text: the

text: other

text: hand,

text: δ

text: is

text: dialed

text: up

text: by

text: a

text: factor

text: of

text: about

text: many

text: pieces

text: of

text: the

text: hexadecapole

text: terms

text: change

text: sign

text: under

text: As

text: a

text: ﬁnal

text: example,

text: we

text: display

text: in

text: Fig.

text: the

text: eﬀect

text: of

text: the

text: transformation

text: ω

text: ω

text: and

text: as

text: a

text: consequence,

text: slightly

text: unequal

text: masses

text: on

text: the

text: generation

text: of

text: orbital

text: ﬂips

text: the

text: initial

text: angle

text: ω

text: yields

text: no

text: ﬂips.

text: via

text: octupole-order

text: terms.

text: We

text: again

text: consider

text: an

text: inner

text: In

text: the

text: foregoing

text: example,

text: the

text: evolutions

text: are

text: the

text: same

text: binary

text: of

text: total

text: mass

text: with

text: a

text: a.u.,

text: orbiting

text: a

text: whether

text: the

text: GR

text: precessions

text: are

text: included

text: or

text: not.

text: We

text: star

text: or

text: black

text: hole

text: of

text: mass

text: at

text: A

text: a.u.

text: The

text: can

text: investigate

text: when

text: GR

text: eﬀects

text: become

text: important

text: by

text: initial

text: conditions

text: are

text: “dialing

text: up”

text: the

text: GR

text: parameter

text: δ

text: while

text: holding

text: α

text: and

text: ﬁxed.

text: This

text: is

text: equivalent

text: either

text: to

text: reducing

text: a

text: and

text: A

text: by

text: the

text: same

text: factor,

text: holding

text: the

text: masses

text: ﬁxed,

text: or

text: to

text: increasing

text: e

text: E

text: z

text: ω

text: ω

text: all

text: the

text: masses

text: by

text: the

text: same

text: factor,

text: holding

text: a

text: and

text: A

text: ﬁxed.

text: We

text: ﬁnd

text: that

text: orbital

text: ﬂips

text: are

text: preserved

text: until

text: δ

text: is

text: about

text: times

text: larger

text: than

text: the

text: value

text: shown

text: in

text: Eq.

text: The

text: equal-mass

text: case

text: shows

text: no

text: orbital

text: ﬂips

text: in

text: this

text: case

text: Another

text: example

text: generates

text: orbital

text: ﬂips

text: from

text: nearly

text: (blue

text: curves

text: in

text: Fig.

text: basically

text: because

text: is

text: coplanar

text: orbits,

text: an

text: analogue

text: of

text: the

text: case

text: discussed

text: in

text: Sec.

text: smaller

text: than

text: in

text: the

text: previous

text: cases,

text: and

text: the

text: hexadecopole

text: The

text: results

text: are

text: shown

text: in

text: Fig.

text: In

text: this

text: case

text: the

text: terms

text: alone

text: are

text: not

text: large

text: enough

text: to

text: do

text: the

text: job.

text: As

text: we

text: )ged(

text: z

text: e

text: )ged(

text: z

text: e

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 W ≡ · and hˆ δa, and we write down the “Lagrange plan- etary equation(cid:114)s” for the evolution of the orbit elements, dY α = Qα(Xβ, Zγ, t) = Aα(Xβ, f)Bα(Zγ, F) , (2.17) dt S dp (cid:114) p3 (cid:20) (cid:21) = 2 , dt Gm 1 + e cos f where the Xβ and Zβ are orbit elements associated de (cid:114)p (cid:20) R 2 cos f + e + e cos2 f S(cid:21) with the inner and outer binaries, respectively, and where = sin f + , the last step recognizes that every term on the right-hand dt Gm 1 + e cos f side can be factorized into a product of terms depending d(cid:36) (cid:114)1 p − R 2 + e cos f S only on one or the other orbital elements and on either = cos f + sin f , dt e Gm 1 + e cos f f or F. Then the average of dY α/dt is approximated as dι (cid:114) p cos(ω + f)W pro(cid:28)duct o(cid:29)f averages of Aα and Bα, in other words = , dt Gm 1 + e cos f dΩ p sin(ω + f)W dY α (cid:104) (cid:105) (cid:104) (cid:90) (cid:105) (cid:90) sin ι = . (2.15) = Aα Bα dt Gm 1 + e cos f dt 1 Pinner 1 Pouter = Aαdt Bαdt , chaTnhgee ianuxpieliraicreyntvearriaanbgllee i(cid:36)s giisvendebﬁyneω˙d=su(cid:36)˙ch− tΩ˙hacots tι.he Pinner 0 Pouter 0 (2.18) (cid:112) (cid:112) For the outer binary, the analogous components of the R ≡ · perturbing acceleration δA, are deﬁned by N δA, S ≡ · W ≡ · 3 where the two orbital periods are given by P = Λ δA and Hˆ δA. The planetary equations inner 3 3 2π a3/Gm and P = 2π A3/GM, with the as- out(cid:28)er for the outer binary take the form of Eqs. (2.15), with → sumption that P P . In integrating over an inner outer suitable replacements of all the relevant variables, p P, → → → orbit of the inner binary, it is useful to convert the an- e E, m M, f F, and so on, and with ω˙ = − 3 gular variable from the true anomaly f, to the eccentric (cid:36)˙ Ω˙ cos ι . 3 3 3 anomaly u, using the relations Combining these equations and inserting the perturb- ing accelerations, it is straightforward to verify directly √ − − that 1 e2 sin u cos u e sin f = − , cos f = − (cid:112) , (2.19) 1 e cos u 1 e cos u d (cid:16) −(cid:112) (cid:17) (Ω Ω ) = 0 , (cid:112) − − dt 3 √ along with r = a(1 e cos u) and dt = a3/Gm(1 d (cid:16) (cid:112) − (cid:17) e cos u)du. For the outer binary, we use the fact that dt = mη Gmp sin ι Mη GMP sin ι = 0 , − dt 3 √ 3 P3/GM(1 + E cos F) 2dF. Thus the orbit averages d may b(cid:28)e writ(cid:29)ten (cid:90) mη Gmp cos ι + Mη GMP cos ι = 0 , (2.16) dt 3 3 − dY α (1 (cid:90)E2)3/2 2π − reﬂecting the conservation of the three components of = Aα(1 e cos u)du dt (2π)2 the total angular momentum. 0 × 2π Bα C. Secular evolution of the orbit elements dF . (2.20) (1 + E cos F)2 0 We now use ﬁrst-order perturbation theory to obtain the secular evolutions of the orbital elements. This is After carrying out the orbital averages, we convert done by substituting constant values of the orbit elements from time t to a dimensionless time scaled by the inner into the right-hand sides of the planetary equations, and orbital period, namely (cid:18) (cid:19) averaging over an orbit of both the inner and outer bina- ries. This is justiﬁed by the fact that the leading order perturbation is at (cid:15)3 and we are going out to order (cid:15)5. ≡ t t Gm 1/2 τ = . (2.21) Were we including terms of dotriocontupole order ((cid:15)6) in P 2π a3 inner the equations of motion, for example, it would have been necessary to invoke second-order perturbation theory for With this scaling, the entire secular dynamics depends the quadrupole terms, in which the full (secular plus peri- on the three dimensionless parameters: odic) solutions at quadrupole order are substituted back into the Lagrange planetary equations and the orbital ≡ m ≡ m m ≡ a average carried out again. α 3 , η 1 2 , (cid:15) . (2.22) Each planetary equation can be written in the generic m m2 A form ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: W

text: and

text: hˆ

text: δa,

text: and

text: we

text: write

text: down

text: the

text: “Lagrange

text: plan-

text: etary

text: for

text: the

text: evolution

text: of

text: the

text: orbit

text: elements,

text: dY

text: α

text: Qα(Xβ,

text: Zγ,

text: t)

text: Aα(Xβ,

text: f)Bα(Zγ,

text: F)

text: dt

text: S

text: dp

text: dt

text: Gm

text: e

text: cos

text: f

text: where

text: the

text: Xβ

text: and

text: Zβ

text: are

text: orbit

text: elements

text: associated

text: de

text: R

text: cos

text: f

text: e

text: e

text: f

text: with

text: the

text: inner

text: and

text: outer

text: binaries,

text: respectively,

text: and

text: where

text: sin

text: f

text: the

text: last

text: step

text: recognizes

text: that

text: every

text: term

text: on

text: the

text: right-hand

text: dt

text: Gm

text: e

text: cos

text: f

text: side

text: can

text: be

text: factorized

text: into

text: a

text: product

text: of

text: terms

text: depending

text: p

text: R

text: e

text: cos

text: f

text: S

text: only

text: on

text: one

text: or

text: the

text: other

text: orbital

text: elements

text: and

text: on

text: either

text: cos

text: f

text: sin

text: f

text: dt

text: e

text: Gm

text: e

text: cos

text: f

text: f

text: or

text: F.

text: Then

text: the

text: average

text: of

text: dY

text: α/dt

text: is

text: approximated

text: as

text: dι

text: p

text: cos(ω

text: f)W

text: averages

text: of

text: Aα

text: and

text: Bα,

text: in

text: other

text: words

text: dt

text: Gm

text: e

text: cos

text: f

text: dΩ

text: p

text: sin(ω

text: f)W

text: dY

text: α

text: sin

text: ι

text: Aα

text: Bα

text: dt

text: Gm

text: e

text: cos

text: f

text: dt

text: Pinner

text: Pouter

text: Aαdt

text: Bαdt

text: chaTnhgee

text: ianuxpieliraicreyntvearriaanbgllee

text: tΩ˙hacots

text: tι.he

text: Pinner

text: Pouter

text: For

text: the

text: outer

text: binary,

text: the

text: analogous

text: components

text: of

text: the

text: R

text: perturbing

text: acceleration

text: δA,

text: are

text: deﬁned

text: by

text: N

text: δA,

text: S

text: W

text: where

text: the

text: two

text: orbital

text: periods

text: are

text: given

text: by

text: P

text: Λ

text: δA

text: and

text: Hˆ

text: δA.

text: The

text: planetary

text: equations

text: inner

text: 2π

text: and

text: P

text: 2π

text: with

text: the

text: as-

text: for

text: the

text: outer

text: binary

text: take

text: the

text: form

text: of

text: Eqs.

text: with

text: sumption

text: that

text: P

text: P

text: In

text: integrating

text: over

text: an

text: inner

text: outer

text: suitable

text: replacements

text: of

text: all

text: the

text: relevant

text: variables,

text: p

text: P,

text: orbit

text: of

text: the

text: inner

text: binary,

text: it

text: is

text: useful

text: to

text: convert

text: the

text: an-

text: e

text: E,

text: m

text: M,

text: f

text: F,

text: and

text: so

text: on,

text: and

text: with

text: ω˙

text: gular

text: variable

text: from

text: the

text: true

text: anomaly

text: f,

text: to

text: the

text: eccentric

text: Ω˙

text: cos

text: ι

text: anomaly

text: u,

text: using

text: the

text: relations

text: Combining

text: these

text: equations

text: and

text: inserting

text: the

text: perturb-

text: ing

text: accelerations,

text: it

text: is

text: straightforward

text: to

text: verify

text: directly

text: that

text: sin

text: u

text: cos

text: u

text: e

text: sin

text: f

text: cos

text: f

text: e

text: cos

text: u

text: e

text: cos

text: u

text: d

text: (Ω

text: Ω

text: dt

text: along

text: with

text: r

text: e

text: cos

text: u)

text: and

text: dt

text: d

text: e

text: cos

text: u)du.

text: For

text: the

text: outer

text: binary,

text: we

text: use

text: the

text: fact

text: that

text: dt

text: mη

text: Gmp

text: sin

text: ι

text: Mη

text: GMP

text: sin

text: ι

text: dt

text: E

text: cos

text: F)

text: Thus

text: the

text: orbit

text: averages

text: d

text: may

text: mη

text: Gmp

text: cos

text: ι

text: Mη

text: GMP

text: cos

text: ι

text: dt

text: dY

text: α

text: 2π

text: reﬂecting

text: the

text: conservation

text: of

text: the

text: three

text: components

text: of

text: e

text: cos

text: u)du

text: dt

text: the

text: total

text: angular

text: momentum.

text: 2π

text: Bα

text: C.

text: Secular

text: evolution

text: of

text: the

text: orbit

text: elements

text: dF

text: E

text: cos

text: We

text: now

text: use

text: ﬁrst-order

text: perturbation

text: theory

text: to

text: obtain

text: the

text: secular

text: evolutions

text: of

text: the

text: orbital

text: elements.

text: This

text: is

text: After

text: carrying

text: out

text: the

text: orbital

text: averages,

text: we

text: convert

text: done

text: by

text: substituting

text: constant

text: values

text: of

text: the

text: orbit

text: elements

text: from

text: time

text: t

text: to

text: a

text: dimensionless

text: time

text: scaled

text: by

text: the

text: inner

text: into

text: the

text: right-hand

text: sides

text: of

text: the

text: planetary

text: equations,

text: and

text: orbital

text: period,

text: namely

text: averaging

text: over

text: an

text: orbit

text: of

text: both

text: the

text: inner

text: and

text: outer

text: bina-

text: ries.

text: This

text: is

text: justiﬁed

text: by

text: the

text: fact

text: that

text: the

text: leading

text: order

text: perturbation

text: is

text: at

text: and

text: we

text: are

text: going

text: out

text: to

text: order

text: t

text: t

text: Gm

text: τ

text: Were

text: we

text: including

text: terms

text: of

text: dotriocontupole

text: order

text: in

text: P

text: 2π

text: inner

text: the

text: equations

text: of

text: motion,

text: for

text: example,

text: it

text: would

text: have

text: been

text: necessary

text: to

text: invoke

text: second-order

text: perturbation

text: theory

text: for

text: With

text: this

text: scaling,

text: the

text: entire

text: secular

text: dynamics

text: depends

text: the

text: quadrupole

text: terms,

text: in

text: which

text: the

text: full

text: (secular

text: plus

text: peri-

text: on

text: the

text: three

text: dimensionless

text: parameters:

text: odic)

text: solutions

text: at

text: quadrupole

text: order

text: are

text: substituted

text: back

text: into

text: the

text: Lagrange

text: planetary

text: equations

text: and

text: the

text: orbital

text: m

text: m

text: m

text: a

text: average

text: carried

text: out

text: again.

text: α

text: η

text: Each

text: planetary

text: equation

text: can

text: be

text: written

text: in

text: the

text: generic

text: m

text: A

text: form

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##