text: − − − 0 0 0 + − − − 0 0 0 + + + − (b) A block of zeros. Note that this alone (a) A block of , otherwise + , above the does not make ρ(x) ∈ SO(2n + 1) block di- − + ∅ agonal, since this is only a 2h-by-2(n h) diagonal. (Or all , if B = ) block of zeros. Figure 1.  ∈ Q ∈ Q  polynomial χ∅ [Y (X)], together with polynomials ψ , χ [Y (X)] for each block ∈ B B B B [X], such (cid:88)that (cid:89) (cid:88) (cid:89) (cid:89)      − − [p] = ψ (y+y )2g + χ (y )2g (y+)2g . B ij ij B ij ij B∈B[X] (i,j)∈B B∈B[X]∪{∅} (i,j)∈B (i,j)∈/B∪B¯ i<j ± Illustration. Our goal is to use the relations between the c (L ) to express any monomial ∈ 1 ij [p] R as a sum of images of “good” collections of sections deﬁned at the end of Section .While we are working in R, we will illustrate monomials appearing in our argument with (cid:18) (cid:19) (cid:18) (cid:19) ± pictures of the form ρ(x) takes in the vanishing loci of the corresponding sections s (x). ∈ × ij × Though ρ(x) SO(2n + 1), our illustrations will be n n grids, in which we represent 2 2 − − x y + × z w blocks of the form by , the 2 2 blocks of the form − by , and y x w z × − 0 the 2 2 blocks of zeros (in the vanishing loci of (s+(x), s (x)) by . We have chosen ij ij ∀ ∈ to illustrate our argument using blocks B with the property that i < j (i, j) B, since these produce the simplest visualisations; note that this is not an assumption we make in ≥ − − the proof. Lemma 4.17 says we can express any monomial of degree 2gn(n 1) n + 1 as a sum of terms whose corresponding sections have vanishing loci taking one of the forms shown in Figure 1. | | Proof. By induction on X . | | { } For X = 2, without loss of generality take X = 1, 2 . Then Q Q − − [Y (X)] = [y+ , y , y+ , y ]. 12 12 21 21 − − − ∈ Q Since [y+ ] = [y+ ] and [y ] = [ y ] in R, any monomial p [Y (X)] of degree at least − 12 21 12 21 − ∈ Q ≥ − ≥ 4g 1 is equivalent in R to λ(y+ )a(y )b, where λ and a + b 4g 1. So either a 2g ≥ ≥ 12 12 − − ∈ B or b 2g. If a 2g, take χ∅ = (y+ )a 2g(y )b, and ψ = χ = 0 for all B [X]. If ≥ − − 12 12 B B∈ B (cid:88) b 2g, take χ = (y+ )a(y )b 2g, and ψ = χ∅ = 0 for all B [X]. ( ) (1,2) 12 12 B 20

text: (b)

text: A

label: caption

text: block

text: of

text: zeros.

text: Note

text: that

text: this

text: alone

text: (a)

text: A

text: block

text: of

text: otherwise

text: above

text: the

text: does

text: not

text: make

text: ρ(x)

text: block

text: di-

text: agonal,

text: since

text: this

text: is

text: only

text: a

text: h)

text: diagonal.

text: (Or

text: all

text: if

text: B

text: block

text: of

text: zeros.

text: Figure

text: Q

text: Q

text: polynomial

text: χ∅

text: [Y

text: (X)],

text: together

text: with

text: polynomials

text: ψ

text: χ

text: [Y

text: (X)]

text: for

text: each

text: block

text: B

text: B

text: B

text: B

text: [X],

text: such

text: [p]

text: ψ

text: (y+y

text: χ

text: (y

text: B

text: ij

text: ij

text: B

text: ij

text: ij

text: B∈B[X]

text: (i,j)∈B

text: B∈B[X]∪{∅}

text: (i,j)∈B

text: (i,j)∈/B∪B¯

text: i<j

text: Illustration.

text: Our

text: goal

text: is

text: to

text: use

text: the

text: relations

text: between

text: the

text: c

text: (L

text: to

text: express

text: any

text: monomial

text: ij

text: [p]

text: R

text: as

text: a

text: sum

text: of

text: images

text: of

text: “good”

text: collections

text: of

text: sections

text: deﬁned

text: at

text: the

text: end

text: of

text: Section

text: .While

text: we

text: are

text: working

text: in

text: R,

text: we

text: will

text: illustrate

text: monomials

text: appearing

text: in

text: our

text: argument

text: with

text: pictures

text: of

text: the

text: form

text: ρ(x)

text: takes

text: in

text: the

text: vanishing

text: loci

text: of

text: the

text: corresponding

text: sections

text: s

text: (x).

text: ij

text: Though

text: ρ(x)

text: our

text: illustrations

text: will

text: be

text: n

text: n

text: grids,

text: in

text: which

text: we

text: represent

text: x

text: y

text: z

text: w

text: blocks

text: of

text: the

text: form

text: by

text: the

text: blocks

text: of

text: the

text: form

text: by

text: and

text: y

text: x

text: w

text: z

text: the

text: blocks

text: of

text: zeros

text: (in

text: the

text: vanishing

text: loci

text: of

text: (s+(x),

text: s

text: (x))

text: by

text: We

text: have

text: chosen

text: ij

text: ij

text: to

text: illustrate

text: our

text: argument

text: using

text: blocks

text: B

text: with

text: the

text: property

text: that

text: i

text: j

text: (i,

text: j)

text: B,

text: since

text: these

text: produce

text: the

text: simplest

text: visualisations;

text: note

text: that

text: this

text: is

text: not

text: an

text: assumption

text: we

text: make

text: in

text: the

text: proof.

text: Lemma

text: says

text: we

text: can

text: express

text: any

text: monomial

text: of

text: degree

text: n

text: as

text: a

text: sum

text: of

text: terms

text: whose

text: corresponding

text: sections

text: have

text: vanishing

text: loci

text: taking

text: one

text: of

text: the

text: forms

text: shown

text: in

text: Figure

text: Proof.

text: By

text: induction

text: on

text: X

text: For

text: X

text: without

text: loss

text: of

text: generality

text: take

text: X

text: Then

text: Q

text: Q

text: [Y

text: (X)]

text: [y+

text: y

text: y+

text: y

text: Q

text: Since

text: [y+

text: [y+

text: and

text: [y

text: y

text: in

text: R,

text: any

text: monomial

text: p

text: [Y

text: (X)]

text: of

text: degree

text: at

text: least

text: Q

text: is

text: equivalent

text: in

text: R

text: to

text: λ(y+

text: )a(y

text: )b,

text: where

text: λ

text: and

text: a

text: b

text: So

text: either

text: a

text: B

text: or

text: b

text: If

text: a

text: take

text: χ∅

text: (y+

text: )a

text: )b,

text: and

text: ψ

text: χ

text: for

text: all

text: B

text: [X].

text: If

text: B

text: B∈

text: B

text: b

text: take

text: χ

text: (y+

text: )a(y

text: )b

text: and

text: ψ

text: χ∅

text: for

text: all

text: B

text: [X].

text: B


text: ∈ to simplify notation we will also denote elements of T by (θ , θ ). Choose an element t T ± ∈ Z 1 2 ± that is generic in the sense that θ θ / 2π . Recall that the roots of g are (2η , 2η , η + − ± ± 1 2 C 1 2 1 η , η η ). If φ = (η η ) we denote by the one dimensional torus representation 2 1 · 2 ± ± j k (φ) × C → (θ , θ ) z = e i(θj θk)z. Consider∼the line bundles∼L = V (t) V (t)/T = S (t) 1 2∈ (cid:81) ∗ φ ⊗ g T (φ(cid:80)) ∗ g − 2,g for φ Φ(G). Observe that L− = L , and L = L L . Hence c (L ) = c (L ), and φ φ φ+ψ φ ψ 1 φ 1 φ c (L ) = c (L ) + c (L ). In this case, Theorem 1.2 says 1 φ+ψ 1 φ 1 ψ ≥ Proposition 2.1. The product ∈ c (L )kφ vanishes whenever ∈ k 8g. φ Φ(G) 1 φ φ Φ(G) φ As in the case of G = SO(3), and following the strategy for G = SU(n) in [5], we prove this by identifying collections of sections of the L with no common zeros, thus observing φ that the corresponding products of ﬁrst Chern classes c (L ) vanish. We will identify three 1 φ such vanishing products, and then prove that any other monomial of degree at least 8g is ∗ equivalent in H (S (t)) to a combination of those three. 2,g ∈ { | ≤ ≤ } ± Deﬁnition 2.2. For x a , b 1 l g , let s (x) be the section of L ± induced by l l ij ηi ηj the T -equivariant map →C V (t) ± √ g (ηi ηj) (cid:55)→ ∓ − ± ρ (ρ(x)) − − (ρ(x)) + 1((ρ(x)) − (ρ(x)) − ). 2i 1,2j 1 2i,2j 2i,2j 1 2i 1,2j − (Note that s (x) is a section of the trivial bundle L .) ii 0 (cid:18) (cid:19) ± Observe that if s (x) vanishes, then the (i, j)th 2-by-2 block in ρ(x) takes the form ij a b − ∓ ± ; if for any given i, j, both s+(x) and s (x) vanish, then the (i, j)th 2-by-2 b a ij ij block in ρ(x) is zero. ± ± To simplify notation, will will write c := c (L ) ij 1 ij − Lemma 2.3. The monomial z := (c+ c+ c )2g vanishes in H12g(S (t)). 1 11 12 12 2,g   ∈ { } − Proof. Let x  a , b , and consider the sections s+ (x), s+ (x), s (x). If these all vanish,  l l  11 12 12  a b 0 0 c   −   b a 0 0 d  ∗ ∗ ∗ ∗ ∗ ∈ R ∈ then ρ(x) = , for some a, b, c, d . Since ρ(x) SO(5), c = d = 0, ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗      a b 0 0 0   −   b a 0 0 0  ∗ ∗ ∗ ∈ R so in fact ρ(x) = 0 0 , for some a, b with a2 + b2 = 1. If all 6g sections ∗ ∗ ∗ 0 0 ∗ ∗ ∗ 0 0 4 ##LTLine##

label: paragraph

text: to

text: simplify

text: notation

text: we

text: will

text: also

text: denote

text: elements

text: of

text: T

text: by

text: (θ

text: θ

text: Choose

text: an

text: element

text: t

text: T

text: Z

text: that

text: is

text: generic

text: in

text: the

text: sense

text: that

text: θ

text: θ

text: 2π

text: Recall

text: that

text: the

text: roots

text: of

text: g

text: are

text: 2η

text: η

text: C

text: η

text: η

text: η

text: If

text: φ

text: (η

text: η

text: we

text: denote

text: by

text: the

text: one

text: dimensional

text: torus

text: representation

text: j

text: k

text: (φ)

text: C

text: (θ

text: θ

text: z

text: e

text: i(θj

text: θk)z.

text: Consider∼the

text: line

text: bundles∼L

text: V

text: (t)

text: V

text: (t)/T

text: S

text: (t)

text: 2∈

text: φ

text: g

text: T

text: g

text: for

text: φ

text: Φ(G).

text: Observe

text: that

text: L−

text: L

text: and

text: L

text: L

text: L

text: Hence

text: c

text: (L

text: c

text: (L

text: and

text: φ

text: φ

text: φ+ψ

text: φ

text: ψ

text: φ

text: φ

text: c

text: (L

text: c

text: (L

text: c

text: (L

text: In

text: this

text: case,

text: Theorem

text: says

text: φ+ψ

text: φ

text: ψ

text: Proposition

text: The

text: product

text: c

text: (L

text: )kφ

text: vanishes

text: whenever

text: k

text: φ

text: Φ(G)

text: φ

text: φ

text: Φ(G)

text: φ

text: As

text: in

text: the

text: case

text: of

text: G

text: and

text: following

text: the

text: strategy

text: for

text: G

text: SU(n)

text: in

text: we

text: prove

text: this

text: by

text: identifying

text: collections

text: of

text: sections

text: of

text: the

text: L

text: with

text: no

text: common

text: zeros,

text: thus

text: observing

text: φ

text: that

text: the

text: corresponding

text: products

text: of

text: ﬁrst

text: Chern

text: classes

text: c

text: (L

text: vanish.

text: We

text: will

text: identify

text: three

text: φ

text: such

text: vanishing

text: products,

text: and

text: then

text: prove

text: that

text: any

text: other

text: monomial

text: of

text: degree

text: at

text: least

text: is

text: equivalent

text: in

text: H

text: (S

text: (t))

text: to

text: a

text: combination

text: of

text: those

text: three.

text: Deﬁnition

text: For

text: x

text: a

text: b

text: l

text: g

text: let

text: s

text: (x)

text: be

text: the

text: section

text: of

text: L

text: induced

text: by

text: l

text: l

text: ij

text: ηi

text: ηj

text: the

text: T

text: -equivariant

text: map

text: →C

text: V

text: (t)

text: g

text: (ηi

text: ηj)

text: ρ

text: (ρ(x))

text: (ρ(x))

text: (ρ(x))

text: (Note

text: that

text: s

text: (x)

text: is

text: a

text: section

text: of

text: the

text: trivial

text: bundle

text: L

text: ii

text: Observe

text: that

text: if

text: s

text: (x)

text: vanishes,

text: then

text: the

text: (i,

text: j)th

text: block

text: in

text: ρ(x)

text: takes

text: the

text: form

text: ij

text: a

text: b

text: if

text: for

text: any

text: given

text: i,

text: j,

text: both

text: s+(x)

text: and

text: s

text: (x)

text: vanish,

text: then

text: the

text: (i,

text: j)th

text: b

text: a

text: ij

text: ij

text: block

text: in

text: ρ(x)

text: is

text: zero.

text: To

text: simplify

text: notation,

text: will

text: will

text: write

text: c

text: c

text: (L

text: ij

text: ij

text: Lemma

text: The

text: monomial

text: z

text: (c+

text: c+

text: c

text: vanishes

text: in

text: (t)).

text: Proof.

text: Let

text: x

text: a

text: b

text: and

text: consider

text: the

text: sections

text: s+

text: (x),

text: s+

text: (x),

text: s

text: (x).

text: If

text: these

text: all

text: vanish,

text: l

text: l

text: a

text: b

text: c

text: b

text: a

text: d

text: R

text: then

text: ρ(x)

text: for

text: some

text: a,

text: b,

text: c,

text: d

text: Since

text: ρ(x)

text: c

text: d

text: a

text: b

text: b

text: a

text: R

text: so

text: in

text: fact

text: ρ(x)

text: for

text: some

text: a,

text: b

text: with

text: If

text: all

text: sections

text: ##LTLine##


text: 18 (1) g = kh for some k ∈ G with s(k) = 1 and some h ∈ V : (cid:48) In this case, let t (g) = u(h). (2) g = kh for some k ∈ G with s(k) = 1 and some h ∈ V U2 \ V : In this case let E = kV and E = complement of kV U2. 1 2 Since E U ∩ E U = ∅, there is a v ∈ T with v| = (ku)| 1 2 E1 E1 and v| = t| . If there are multiple choices for v, choose the E2 E2 lexicographically least conﬁguration for a ﬁxed ordering of G (cid:48) and a ﬁxed ordering of A. Let t (g) = v(kh). (3) g (cid:54)= kh for s(k) = 1 and h ∈ V U2: (cid:48) In this case, let t (g) = t(g). Since the 1’s in S are Z-apart, this leads to a well-deﬁned deﬁnition (cid:48) (cid:48) for t . Informally, t is constructed from t as follows: the conﬁguration (cid:48) t mostly agrees with t. The ﬁrst exceptions are the V -neighborhoods (cid:48) of any k ∈ G such that s(k) = 1, where we set t to equal the pattern that appears around the origin in u. The second exceptions are the borders of these V -neighborhoods, where some adjustments need to be (cid:48) made so that—as we explain below—t and t agree on any translate of X. This construction is schematically depicted in Figure 3. The following hold: (cid:48) • φ is continuous and equivariant. So T = φ(S ×T ) is a subshift. • Since strong irreducibility is closed under taking products and (cid:48) factors, we see that T is strongly irreducible. (cid:48) (cid:48) (cid:48) − • Let t = φ(s , t ), t = φ(s , t ) ∈ T . Since S is a (V U2X)(V U2X) 1- 1 1 1 2 2 2 (cid:48) witness shift, there is a g ∈ G with s (g) = s (g) = 1. So, t | (cid:48) 1 2 − (cid:48) 1 gV and t | are both translates of u| , which means (g 1t )| = − (cid:48)2 gV V − (cid:48) −1 (cid:48)V (g 1t )| . So from the deﬁnition of V , we get d(g 1t , g 1t ) < 2 V (cid:48) 1 2 (cid:15). Hence T is (cid:15)-proximal. (cid:48) • Now we claim that the set of X-patterns of T and T are equal. First notice that since u| has all the X-patterns in T , and (cid:48) V u| appears in T . So we get that all the X-patterns of T appear V (cid:48) in T . (cid:48) (cid:48) (cid:48) Now let t = φ(s, t) ∈ T and ﬁx an X-pattern in t , located at gX. If gX does not meet any k(V U2X) for s(k) = 1, then (cid:48) t | = t| and so the pattern appears in T . If, on the other gX gX hand, gX intersects k(V U2X) for some k with s(k) = 1, by the (cid:48) deﬁnition of t around k, we again see that the pattern in gX appears in T . (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) • To see ε-minimality of T , let t , t ∈ T . We know that t | 1 2 2 X(cid:48) is one of the X-patterns in T , so it appears somewhere in t , (cid:48) (cid:48) 1 i.e. there exists a g ∈ G such that (gt )| agrees with t | . 1 X 2 X

label: list

text: g

text: kh

text: for

text: some

text: k

text: G

text: with

text: s(k)

text: and

text: some

text: h

text: V

text: In

text: this

text: case,

text: let

text: t

text: (g)

text: u(h).

text: g

text: kh

text: for

text: some

text: k

text: G

text: with

text: s(k)

text: and

text: some

text: h

text: V

text: V

text: In

text: this

text: case

text: let

text: E

text: kV

text: and

text: E

text: complement

text: of

text: kV

text: Since

text: E

text: U

text: E

text: U

text: there

text: is

text: a

text: v

text: T

text: with

text: v|

text: (ku)|

text: and

text: v|

text: t|

text: If

text: there

text: are

text: multiple

text: choices

text: for

text: v,

text: choose

text: the

text: lexicographically

text: least

text: conﬁguration

text: for

text: a

text: ﬁxed

text: ordering

text: of

text: G

text: and

text: a

text: ﬁxed

text: ordering

text: of

text: A.

text: Let

text: t

text: (g)

text: v(kh).

text: g

text: kh

text: for

text: s(k)

text: and

text: h

text: V

text: In

text: this

text: case,

text: let

text: t

text: (g)

text: t(g).

text: Since

text: the

text: in

text: S

text: are

text: Z-apart,

text: this

text: leads

text: to

text: a

text: well-deﬁned

text: deﬁnition

text: for

text: t

text: Informally,

text: t

text: is

text: constructed

text: from

text: t

text: as

text: follows:

text: the

text: conﬁguration

text: t

text: mostly

text: agrees

text: with

text: t.

text: The

text: ﬁrst

text: exceptions

text: are

text: the

text: V

text: -neighborhoods

text: of

text: any

text: k

text: G

text: such

text: that

text: s(k)

text: where

text: we

text: set

text: t

text: to

text: equal

text: the

text: pattern

text: that

text: appears

text: around

text: the

text: origin

text: in

text: u.

text: The

text: second

text: exceptions

text: are

text: the

text: borders

text: of

text: these

text: V

text: -neighborhoods,

text: where

text: some

text: adjustments

text: need

text: to

text: be

text: made

text: so

text: that—as

text: we

text: explain

text: below—t

text: and

text: t

text: agree

text: on

text: any

text: translate

text: of

text: X.

text: This

text: construction

text: is

text: schematically

text: depicted

text: in

text: Figure

text: The

text: following

text: hold:

text: φ

text: is

text: continuous

text: and

text: equivariant.

text: So

text: T

text: φ(S

text: ×T

text: is

text: a

text: subshift.

text: Since

text: strong

text: irreducibility

text: is

text: closed

text: under

text: taking

text: products

text: and

text: factors,

text: we

text: see

text: that

text: T

text: is

text: strongly

text: irreducible.

text: Let

text: t

text: φ(s

text: t

text: t

text: φ(s

text: t

text: T

text: Since

text: S

text: is

text: a

text: (V

text: witness

text: shift,

text: there

text: is

text: a

text: g

text: G

text: with

text: s

text: (g)

text: s

text: (g)

text: So,

text: t

text: gV

text: and

text: t

text: are

text: both

text: translates

text: of

text: u|

text: which

text: means

text: (g

text: gV

text: V

text: −1

text: (g

text: So

text: from

text: the

text: deﬁnition

text: of

text: V

text: we

text: get

text: d(g

text: g

text: V

text: Hence

text: T

text: is

text: Now

text: we

text: claim

text: that

text: the

text: set

text: of

text: X-patterns

text: of

text: T

text: and

text: T

text: are

text: equal.

text: First

text: notice

text: that

text: since

text: u|

text: has

text: all

text: the

text: X-patterns

text: in

text: T

text: and

text: V

text: u|

text: appears

text: in

text: T

text: So

text: we

text: get

text: that

text: all

text: the

text: X-patterns

text: of

text: T

text: appear

text: V

text: in

text: T

text: Now

text: let

text: t

text: φ(s,

text: t)

text: T

text: and

text: ﬁx

text: an

text: X-pattern

text: in

text: t

text: located

text: at

text: gX.

text: If

text: gX

text: does

text: not

text: meet

text: any

text: k(V

text: for

text: s(k)

text: then

text: t

text: t|

text: and

text: so

text: the

text: pattern

text: appears

text: in

text: T

text: If,

text: on

text: the

text: other

text: gX

text: gX

text: hand,

text: gX

text: intersects

text: k(V

text: for

text: some

text: k

text: with

text: s(k)

text: by

text: the

text: deﬁnition

text: of

text: t

text: around

text: k,

text: we

text: again

text: see

text: that

text: the

text: pattern

text: in

text: gX

text: appears

text: in

text: T

text: To

text: see

text: ε-minimality

text: of

text: T

text: let

text: t

text: t

text: T

text: We

text: know

text: that

text: t

text: is

text: one

text: of

text: the

text: X-patterns

text: in

text: T

text: so

text: it

text: appears

text: somewhere

text: in

text: t

text: i.e.

text: there

text: exists

text: a

text: g

text: G

text: such

text: that

text: (gt

text: agrees

text: with

text: t

text: X

text: X


text: 19 k3(VU2 \V) k3VU2X k1(VU2 \ V) k1VU2X k3V k1V k3 k1 k2(VU2 \V) k2VU2X k2V k2 ki’s are points g ∈ G for which s(g) = 1. The restriction of t′ to kiV equals a translate of the restriction of u to V . In ki(V U2 \ V ), t′ is ﬁlled in a way that t′ locally looks like an alement of T around ki t′ agrees with t outside of kiV U2’s. (cid:48) Figure 3. t = φ(s, t) for s ∈ S and t ∈ T . We assumed that any two conﬁgurations that agree on X have (cid:48) (cid:48) distance less than ε. So, d(gt , t ) < ε. 1 2 This concludes the proof of Proposition 2.11. First we prove proximal shifts are a G . Given a shift S, an ε > 0 δ and a g ∈ G, let P ⊂ S × S be the set of pairs of conﬁgurations s , s g 1 2 such that d(gs , gs ) < ε. Since P is the preimage of an open set 1 2 g under a continuous map P is open. Thus, whenever S is ε-proximal g the collection {P : g ∈ G} forms an open cover of S × S and thus, g by compactness, whenever a shift is ε-proximal there is a ﬁnite subset X ⊂ G which suﬃce to demonstrate this. For each X ⊂ G, whether X demonstrates ε-proximality is determined by the restriction of S to a ﬁnite set of elements of G. But this is exactly the deﬁnition of a clopen set in the topology on the space of shifts. Thus the set of ε-proximal shifts is the union of a collection of clopen sets and is therefore open. Now we prove minimal shifts are a G . To do this, since minimal δ shifts are exactly the shifts that are ε-minimal for all ε > 0, it is enough ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

font: default

text: \V)

text: V)

text: \V)

text: ki’s

text: are

text: points

text: g

text: G

text: for

text: which

text: s(g)

text: The

text: restriction

text: of

text: t′

text: to

text: kiV

text: equals

text: a

text: translate

text: of

text: the

text: restriction

text: of

text: u

text: to

text: V

text: In

text: ki(V

text: V

text: t′

text: is

text: ﬁlled

text: in

text: a

text: way

text: that

text: t′

text: locally

text: looks

text: like

text: an

text: alement

text: of

text: T

text: around

text: ki

text: t′

text: agrees

text: with

text: t

text: outside

text: of

text: kiV

text: Figure

text: t

text: φ(s,

text: t)

text: for

text: s

text: S

text: and

text: t

text: T

text: We

text: assumed

text: that

text: any

text: two

text: conﬁgurations

text: that

text: agree

text: on

text: X

text: have

text: distance

text: less

text: than

text: ε.

text: So,

text: d(gt

text: t

text: ε.

text: This

text: concludes

text: the

text: proof

text: of

text: Proposition

text: First

text: we

text: prove

text: proximal

text: shifts

text: are

text: a

text: G

text: Given

text: a

text: shift

text: S,

text: an

text: ε

text: δ

text: and

text: a

text: g

text: G,

text: let

text: P

text: S

text: S

text: be

text: the

text: set

text: of

text: pairs

text: of

text: conﬁgurations

text: s

text: s

text: g

text: such

text: that

text: d(gs

text: gs

text: ε.

text: Since

text: P

text: is

text: the

text: preimage

text: of

text: an

text: open

text: set

text: g

text: under

text: a

text: continuous

text: map

text: P

text: is

text: open.

text: Thus,

text: whenever

text: S

text: is

text: ε-proximal

text: g

text: the

text: collection

text: {P

text: g

text: G}

text: forms

text: an

text: open

text: cover

text: of

text: S

text: S

text: and

text: thus,

text: g

text: by

text: compactness,

text: whenever

text: a

text: shift

text: is

text: ε-proximal

text: there

text: is

text: a

text: ﬁnite

text: subset

text: X

text: G

text: which

text: suﬃce

text: to

text: demonstrate

text: this.

text: For

text: each

text: X

text: G,

text: whether

text: X

text: demonstrates

text: ε-proximality

text: is

text: determined

text: by

text: the

text: restriction

text: of

text: S

text: to

text: a

text: ﬁnite

text: set

text: of

text: elements

text: of

text: G.

text: But

text: this

text: is

text: exactly

text: the

text: deﬁnition

text: of

text: a

text: clopen

text: set

text: in

text: the

text: topology

text: on

text: the

text: space

text: of

text: shifts.

text: Thus

text: the

text: set

text: of

text: ε-proximal

text: shifts

text: is

text: the

text: union

text: of

text: a

text: collection

text: of

text: clopen

text: sets

text: and

text: is

text: therefore

text: open.

text: Now

text: we

text: prove

text: minimal

text: shifts

text: are

text: a

text: G

text: To

text: do

text: this,

text: since

text: minimal

text: δ

text: shifts

text: are

text: exactly

text: the

text: shifts

text: that

text: are

text: ε-minimal

text: for

text: all

text: ε

text: it

text: is

text: enough

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 21 [8] Eli GOn the FC-chain of a grouplasner and Benjamin Weiss, Minimal actions of the group of permutations of the integers, Geometric and Functional Analysis 12 (2002), no. 5, 964–988. [9] Franklin Haimo, On the FC-chain of a group, Canadian Journal of Mathemat- ics 5 (1953), 498–511. [10] Yair Hartman, Kate Juschenko, Omer Tamuz, and Pooya Vahidi Ferdowsi, Thompson’s group F is not strongly amenable, Ergodic Theory and Dynamical Systems (2017). [11] Mehrdad Kalantar and Matthew Kennedy, Boundaries of reduced C∗-algebras of discrete groups, Journal fu¨r die reine und angewandte Mathematik (Crelles Journal) 727 (2017), 247–267. [12] DH McLain, Remarks on the upper central series of a group, Glasgow Mathe- matical Journal 3 (1956), no. 1, 38–44. [13] Bernhard H Neumann, Groups covered by permutable subsets, Journal of the London Mathematical Society 1 (1954), no. 2, 236–248. California Institute of Technology

label: reference

text: Eli

text: GOn

text: the

text: FC-chain

text: of

text: a

text: grouplasner

text: and

text: Benjamin

text: Weiss,

text: Minimal

text: actions

text: of

text: the

text: group

text: of

text: permutations

text: of

text: the

text: integers,

text: Geometric

text: and

text: Functional

text: Analysis

text: no.

text: Franklin

text: Haimo,

text: On

text: the

text: FC-chain

text: of

text: a

text: group,

text: Canadian

text: Journal

text: of

text: Mathemat-

text: ics

text: Yair

text: Hartman,

text: Kate

text: Juschenko,

text: Omer

text: Tamuz,

text: and

text: Pooya

text: Vahidi

text: Ferdowsi,

text: Thompson’s

text: group

text: F

text: is

text: not

text: strongly

text: amenable,

text: Ergodic

text: Theory

text: and

text: Dynamical

text: Systems

text: Mehrdad

text: Kalantar

text: and

text: Matthew

text: Kennedy,

text: Boundaries

text: of

text: reduced

text: C∗-algebras

text: of

text: discrete

text: groups,

text: Journal

text: fu¨r

text: die

text: reine

text: und

text: angewandte

text: Mathematik

text: (Crelles

text: Journal)

text: DH

text: McLain,

text: Remarks

text: on

text: the

text: upper

text: central

text: series

text: of

text: a

text: group,

text: Glasgow

text: Mathe-

text: matical

text: Journal

text: no.

text: Bernhard

text: H

text: Neumann,

text: Groups

text: covered

text: by

text: permutable

text: subsets,

text: Journal

text: of

text: the

text: London

text: Mathematical

text: Society

text: no.

text: California

text: Institute

text: of

text: Technology


text: RELATIONAL CHARACTERISATIONS OF PATHS 17 that connects x with y. This path W together with the edge from y to x gives the required cycle in Line 22. In the ﬁrst step, the directed tree D is constructed as follows. Line 7 initialises the relation D as the empty tree. The while-loop maintains the vector v that contains all vertices of the tree. At the start of the loop, v contains only the point x which is the root of D; see Line 8. As long as the vertex y has not been reached, which is checked in Line 9, the algorithm chooses an edge e of R that goes from a vertex in v to a vertex outside v in Line 10. The edge is added to the tree in Line 11 and its end point is added to v in Line 12. An edge is a relation consisting of a single pair and can therefore be represented by an atom. We use the predicate atom(a) to specify that the relation a is an atom in the relation-algebraic sense as deﬁned in Section . Similarly to points, we use an operation chooseAtom(x) that (deterministically) chooses an atom contained in a non-empty relation x. To reason about the operation chooseAtom, we assume it satisﬁes the following axioms: ⊆ (cid:54) ⇒ chooseAtom(x) x x = O atom(chooseAtom(x)) Since the while-loop adds edges leaving the set v and since it is known that y is reachable from x, the vertex y will eventually be included in v, so D will contain a path from x to y. ∗ ⊆ ∗ By construction, D is acyclic and satisﬁes D ; y DT ; x. As a consequence we can use Algorithm 1 to determine a path from x to y; Lines 14–21 of Algorithm 3 are identical to Lines 2–9 of Algorithm 1. When the second while-loop of Algorithm 3 terminates, W contains a terminating path from x to y. Line 22 adds to this path the edge from y to x to obtain a cycle C. By the choice of x and y in Lines 2 and 3 this edge is contained in R. We now discuss the postcondition of Algorithm 3. It constructs a non-empty cycle C that is also a subgraph of R. These conditions immediately translate into the following formulas, whose conjunction is the postcondition Post(R, C): (cid:54) ⊆ C = O cycle(C) C R Here the condition cycle(C) speciﬁes that C is a cycle as deﬁned in Section . We conclude this section by discussing the invariant used to prove partial correctness of Algorithm 3 with respect to Pre(R) and Post(R, C). As before, Isabelle/HOL’s Hoare logic tactic splits the correctness proof into a number of veriﬁcation conditions. For example, the if-statement is split into two goals, one for each case. For the case x = y the postcondition is easy to establish since the loop on this vertex is immediately available. (cid:54) The case x = y requires more eﬀort. To reuse Algorithm 1 we have to show that its precondition Pre(D, x, y) holds, which is the conjunction of the following formulas: ⊆ (cid:54) ∗ ⊆ ∗ D+ I point(x) point(y) x = y D ; y DT ; x To establish these conditions we use the conjunction of the following formulas as the invariant for the ﬁrst while-loop in Lines 9–13: (cid:54) ⊆ ∗ ⊆ point(x) point(y) x = y y RT ; x y ; xT R ⊆ ⊆ ⊆ ⊆ D R D+ I D ; DT I D v ; vT v = v ; L ⊆ ∗ ∪ x ; vT D D ; x = O v = x DT ; L The ﬁrst line lists simple conditions that follow immediately from the selection of x and y in Lines 2 and 3 of Algorithm 3. These properties are needed later and have to be maintained throughout the entire proof, which is not diﬃcult since R, x and y are not modiﬁed in Lines 4–23. ##LTLine## ##LTLine##

text: RELATIONAL

text: CHARACTERISATIONS

text: OF

text: PATHS

text: that

text: connects

text: x

text: with

text: y.

text: This

text: path

text: W

text: together

text: with

text: the

text: edge

text: from

text: y

text: to

text: x

text: gives

text: the

text: required

text: cycle

text: in

text: Line

text: In

text: the

text: ﬁrst

text: step,

text: the

text: directed

text: tree

text: D

text: is

text: constructed

text: as

text: follows.

text: Line

text: initialises

text: the

text: relation

text: D

text: as

text: the

text: empty

text: tree.

text: The

text: while-loop

text: maintains

text: the

text: vector

text: v

text: that

text: contains

text: all

text: vertices

text: of

text: the

text: tree.

text: At

text: the

text: start

text: of

text: the

text: loop,

text: v

text: contains

text: only

text: the

text: point

text: x

text: which

text: is

text: the

text: root

text: of

text: D;

text: see

text: Line

text: As

text: long

text: as

text: the

text: vertex

text: y

text: has

text: not

text: been

text: reached,

text: which

text: is

text: checked

text: in

text: Line

text: the

text: algorithm

text: chooses

text: an

text: edge

text: e

text: of

text: R

text: that

text: goes

text: from

text: a

text: vertex

text: in

text: v

text: to

text: a

text: vertex

text: outside

text: v

text: in

text: Line

text: The

text: edge

text: is

text: added

text: to

text: the

text: tree

text: in

text: Line

text: and

text: its

text: end

text: point

text: is

text: added

text: to

text: v

text: in

text: Line

text: An

text: edge

text: is

text: a

text: relation

text: consisting

text: of

text: a

text: single

text: pair

text: and

text: can

text: therefore

text: be

text: represented

text: by

text: an

text: atom.

text: We

text: use

text: the

text: predicate

text: atom(a)

text: to

text: specify

text: that

text: the

text: relation

text: a

text: is

text: an

text: atom

text: in

text: the

text: relation-algebraic

text: sense

text: as

text: deﬁned

text: in

text: Section

text: Similarly

text: to

text: points,

text: we

text: use

text: an

text: operation

text: chooseAtom(x)

text: that

text: (deterministically)

text: chooses

text: an

text: atom

text: contained

text: in

text: a

text: non-empty

text: relation

text: x.

text: To

text: reason

text: about

text: the

text: operation

text: chooseAtom,

text: we

text: assume

text: it

text: satisﬁes

text: the

text: following

text: axioms:

text: chooseAtom(x)

text: x

text: x

text: O

text: atom(chooseAtom(x))

text: Since

text: the

text: while-loop

text: adds

text: edges

text: leaving

text: the

text: set

text: v

text: and

text: since

text: it

text: is

text: known

text: that

text: y

text: is

text: reachable

text: from

text: x,

text: the

text: vertex

text: y

text: will

text: eventually

text: be

text: included

text: in

text: v,

text: so

text: D

text: will

text: contain

text: a

text: path

text: from

text: x

text: to

text: y.

text: By

text: construction,

text: D

text: is

text: acyclic

text: and

text: satisﬁes

text: D

text: y

text: DT

text: x.

text: As

text: a

text: consequence

text: we

text: can

text: use

text: Algorithm

text: to

text: determine

text: a

text: path

text: from

text: x

text: to

text: y;

text: Lines

text: of

text: Algorithm

text: are

text: identical

text: to

text: Lines

text: of

text: Algorithm

text: When

text: the

text: second

text: while-loop

text: of

text: Algorithm

text: terminates,

text: W

text: contains

text: a

text: terminating

text: path

text: from

text: x

text: to

text: y.

text: Line

text: adds

text: to

text: this

text: path

text: the

text: edge

text: from

text: y

text: to

text: x

text: to

text: obtain

text: a

text: cycle

text: C.

text: By

text: the

text: choice

text: of

text: x

text: and

text: y

text: in

text: Lines

text: and

text: this

text: edge

text: is

text: contained

text: in

text: R.

text: We

text: now

text: discuss

text: the

text: postcondition

text: of

text: Algorithm

text: It

text: constructs

text: a

text: non-empty

text: cycle

text: C

text: that

text: is

text: also

text: a

text: subgraph

text: of

text: R.

text: These

text: conditions

text: immediately

text: translate

text: into

text: the

text: following

text: formulas,

text: whose

text: conjunction

text: is

text: the

text: postcondition

text: Post(R,

text: C):

text: C

text: O

text: cycle(C)

text: C

text: R

text: Here

text: the

text: condition

text: cycle(C)

text: speciﬁes

text: that

text: C

text: is

text: a

text: cycle

text: as

text: deﬁned

text: in

text: Section

text: We

text: conclude

text: this

text: section

text: by

text: discussing

text: the

text: invariant

text: used

text: to

text: prove

text: partial

text: correctness

text: of

text: Algorithm

text: with

text: respect

text: to

text: Pre(R)

text: and

text: Post(R,

text: C).

text: As

text: before,

text: Isabelle/HOL’s

text: Hoare

text: logic

text: tactic

text: splits

text: the

text: correctness

text: proof

text: into

text: a

text: number

text: of

text: veriﬁcation

text: conditions.

text: For

text: example,

text: the

text: if-statement

text: is

text: split

text: into

text: two

text: goals,

text: one

text: for

text: each

text: case.

text: For

text: the

text: case

text: x

text: y

text: the

text: postcondition

text: is

text: easy

text: to

text: establish

text: since

text: the

text: loop

text: on

text: this

text: vertex

text: is

text: immediately

text: available.

text: The

text: case

text: x

text: y

text: requires

text: more

text: eﬀort.

text: To

text: reuse

text: Algorithm

text: we

text: have

text: to

text: show

text: that

text: its

text: precondition

text: Pre(D,

text: x,

text: y)

text: holds,

text: which

text: is

text: the

text: conjunction

text: of

text: the

text: following

text: formulas:

text: D+

text: I

text: point(x)

text: point(y)

text: x

text: y

text: D

text: y

text: DT

text: x

text: To

text: establish

text: these

text: conditions

text: we

text: use

text: the

text: conjunction

text: of

text: the

text: following

text: formulas

text: as

text: the

text: invariant

text: for

text: the

text: ﬁrst

text: while-loop

text: in

text: Lines

text: point(x)

text: point(y)

text: x

text: y

text: y

text: RT

text: x

text: y

text: xT

text: R

text: D

text: R

text: D+

text: I

text: D

text: DT

text: I

text: D

text: v

text: vT

text: v

text: v

text: L

text: x

text: vT

text: D

text: D

text: x

text: O

text: v

text: x

text: DT

text: L

text: The

text: ﬁrst

text: line

text: lists

text: simple

text: conditions

text: that

text: follow

text: immediately

text: from

text: the

text: selection

text: of

text: x

text: and

text: y

text: in

text: Lines

text: and

text: of

text: Algorithm

text: These

text: properties

text: are

text: needed

text: later

text: and

text: have

text: to

text: be

text: maintained

text: throughout

text: the

text: entire

text: proof,

text: which

text: is

text: not

text: diﬃcult

text: since

text: R,

text: x

text: and

text: y

text: are

text: not

text: modiﬁed

text: in

text: Lines

text: ##LTLine##

text: ##LTLine##


text: 24 R. BERGHAMMER, H. FURUSAWA, W. GUTTMANN, AND P. HO¨FNER [57] S. Schulz. System Description: E 1.8. In K. McMillan, A. Middeldorp, and A. Voronkov, editors, Logic for Programming Artiﬁcial Intelligence and Reasoning (LPAR 19), volume 8312 of Lecture Notes in Computer Science. Springer, 2013. [58] G. Scollo, G. Franco, and V. Manca. A relational view of recurrence and attractors in state transition dynamics. In R. A. Schmidt, editor, Relations and Kleene Algebra in Computer Science, volume 4136 of Lecture Notes in Computer Science, pages 358–372. Springer, 2006. [59] A. Tarski. On the calculus of relations. The Journal of Symbolic Logic, 6(3):73–89, 1941. [60] G. Tinhofer. Methoden der angewandten Graphentheorie. Springer, 1976. [61] J. von Wright. Towards a reﬁnement algebra. Science of Computer Programming, 51(1–2):23–45, 2004. [62] M. Wenzel. Isabelle, Isar – A Versatile Environment for Human Readable Formal Proof Documents. PhD thesis, Technical University Munich, Germany, 2002.

text: R.

text: BERGHAMMER,

text: H.

text: FURUSAWA,

text: W.

text: GUTTMANN,

text: AND

text: P.

text: HO¨FNER

text: S.

text: Schulz.

text: System

text: Description:

text: E

text: In

text: K.

text: McMillan,

text: A.

text: Middeldorp,

text: and

text: A.

text: Voronkov,

text: editors,

text: Logic

text: for

text: Programming

text: Artiﬁcial

text: Intelligence

text: and

text: Reasoning

text: (LPAR

text: volume

text: of

text: Lecture

text: Notes

text: in

text: Computer

text: Science.

text: Springer,

text: G.

text: Scollo,

text: G.

text: Franco,

text: and

text: V.

text: Manca.

text: A

text: relational

text: view

text: of

text: recurrence

text: and

text: attractors

text: in

text: state

text: transition

text: dynamics.

text: In

text: R.

text: A.

text: Schmidt,

text: editor,

text: Relations

text: and

text: Kleene

text: Algebra

text: in

text: Computer

text: Science,

text: volume

text: of

text: Lecture

text: Notes

text: in

text: Computer

text: Science,

text: pages

text: Springer,

text: A.

text: Tarski.

text: On

text: the

text: calculus

text: of

text: relations.

text: The

text: Journal

text: of

text: Symbolic

text: Logic,

text: G.

text: Tinhofer.

text: Methoden

text: der

text: angewandten

text: Graphentheorie.

text: Springer,

text: J.

text: von

text: Wright.

text: Towards

text: a

text: reﬁnement

text: algebra.

text: Science

text: of

text: Computer

text: Programming,

text: M.

text: Wenzel.

text: Isabelle,

text: Isar

text: A

text: Versatile

text: Environment

text: for

text: Human

text: Readable

text: Formal

text: Proof

text: Documents.

text: PhD

text: thesis,

text: Technical

text: University

text: Munich,

text: Germany,


text: A LOWER BOUND FOR THE DOUBLE SLICE GENUS 11 In this subsection we prove Theorem 1.5, which depends on the following three nontrivial results. The ﬁrst one is due to Freedman. Theorem 3.3 (Theorem 1.13 of [5]). Let K be a knot in S3 such that ∆ (t) = 1, ⊂ K then K−bou∼ndZs a locally ﬂat, topologically embedded disk D D4 such that π (D4 D) = . 1 The second result is the key Proposition proved in [3], which is utilized in con- junction with the previous theorem to show the degree of the Alexander polynomial is an upper bound for the topological slice genus. Proposition 3.4 ([3]). Let K be a knot. Every Seifert surface F of K contains a separating simple closed curve L such that: (i) The Alexander polynomial of L is trivial. − (ii) The connected component C of F L that does not contain K is a Seifert − surface for L with g(C) = g(F) deg(∆ (t)). K Remark. In [3] Feller uses deg(∆ (t)) to stand for the breadth of of ∆ (t), while K K we use this notation to stand for the highest degree of the symmetric Alexander polynomial. The third result we need is the unknotting theorem proved by Hillman and Kawauchi, which characterizes when a surface bounds a handbody in S4 in terms of the fundamental group of the surface complement. S Theorem 3S.5 ([10]). Let be a locally ﬂat, topologically embe−ddSed ∼cloZsed surface in S4. Then bounds a handlebody in S4 if and only if π (S4 ) = . 1 With these preparations at hand, we are ready to prove Theorem 1.5. Proof of Theorem 1.5. Start with a Seifert surface F for K, and let L and C be × ∪ as in Proposition 3.4. Treat D4 as S3 [0, 1] ×{ } B4, where B4 is just the S3 1 four ball with diﬀerent notation to avoid confusion. Construct a surface Fˆ by − × paπnu(dsBhCi4n−gistCˆhp)eu∼=sihneZtde.riBionyrtooTfBhFe4o,ridenmetnoo3tD.e3d4, LsbuybcohCˆu.tnhdIasttai(sFdeisaksyDCt)owihsseopesueπsc1ho(eDmdp4iln−etmoFˆeSn)t3∼=alZso[0a,hn1a]ds, 1 cthyecnlicgf(uF˜n)da=mdeengt(a∆l gr(otu)p).. BLeytVF˜anbeKoabmtpaienne’ds tfrhoemoreFˆm,byonreepsleaecsinπg (CˆD4in−BF˜4)b∼=y DZ,. K 1 Again by Van Kampen’s theorem we see the double of F˜ has complement with cyclic fundamental group, and hence by Theorem 3.5 we know it bounds a handlebody. ≤ (cid:3) Therefore, gs (K) deg(∆ (t)). top K As we mentioned in the introduction, this theorem has the following immediate corollaries. The ﬁrst one is Corollary 1.6, which says deg(∆ (t)) = 1 implies K gs (K) = 1. top

text: A

text: LOWER

text: BOUND

text: FOR

text: THE

text: DOUBLE

text: SLICE

text: GENUS

text: In

text: this

text: subsection

text: we

text: prove

text: Theorem

text: which

text: depends

text: on

text: the

text: following

text: three

text: nontrivial

text: results.

text: The

text: ﬁrst

text: one

text: is

text: due

text: to

text: Freedman.

text: Theorem

text: (Theorem

text: of

text: Let

text: K

text: be

text: a

text: knot

text: in

text: such

text: that

text: (t)

text: K

text: then

text: K−bou∼ndZs

text: a

text: locally

text: ﬂat,

text: topologically

text: embedded

text: disk

text: D

text: such

text: that

text: π

text: D)

text: The

text: second

text: result

text: is

text: the

text: key

text: Proposition

text: proved

text: in

text: which

text: is

text: utilized

text: in

text: con-

text: junction

text: with

text: the

text: previous

text: theorem

text: to

text: show

text: the

text: degree

text: of

text: the

text: Alexander

text: polynomial

text: is

text: an

text: upper

text: bound

text: for

text: the

text: topological

text: slice

text: genus.

text: Proposition

text: Let

text: K

text: be

text: a

text: knot.

text: Every

text: Seifert

text: surface

text: F

text: of

text: K

text: contains

text: a

text: separating

text: simple

text: closed

text: curve

text: L

text: such

text: that:

text: (i)

text: The

text: Alexander

text: polynomial

text: of

text: L

text: is

text: trivial.

text: (ii)

text: The

text: connected

text: component

text: C

text: of

text: F

text: L

text: that

text: does

text: not

text: contain

text: K

text: is

text: a

text: Seifert

text: surface

text: for

text: L

text: with

text: g(C)

text: g(F)

text: deg(∆

text: (t)).

text: K

text: Remark.

text: In

text: Feller

text: uses

text: deg(∆

text: (t))

text: to

text: stand

text: for

text: the

text: breadth

text: of

text: of

text: (t),

text: while

text: K

text: K

text: we

text: use

text: this

text: notation

text: to

text: stand

text: for

text: the

text: highest

text: degree

text: of

text: the

text: symmetric

text: Alexander

text: polynomial.

text: The

text: third

text: result

text: we

text: need

text: is

text: the

text: unknotting

text: theorem

text: proved

text: by

text: Hillman

text: and

text: Kawauchi,

text: which

text: characterizes

text: when

text: a

text: surface

text: bounds

text: a

text: handbody

text: in

text: in

text: terms

text: of

text: the

text: fundamental

text: group

text: of

text: the

text: surface

text: complement.

text: S

text: Theorem

text: Let

text: be

text: a

text: locally

text: ﬂat,

text: topologically

text: embe−ddSed

text: ∼cloZsed

text: surface

text: in

text: Then

text: bounds

text: a

text: handlebody

text: in

text: if

text: and

text: only

text: if

text: π

text: With

text: these

text: preparations

text: at

text: hand,

text: we

text: are

text: ready

text: to

text: prove

text: Theorem

text: Proof

text: of

text: Theorem

text: Start

text: with

text: a

text: Seifert

text: surface

text: F

text: for

text: K,

text: and

text: let

text: L

text: and

text: C

text: be

text: as

text: in

text: Proposition

text: Treat

text: as

text: where

text: is

text: just

text: the

text: four

text: ball

text: with

text: diﬀerent

text: notation

text: to

text: avoid

text: confusion.

text: Construct

text: a

text: surface

text: Fˆ

text: by

text: cthyecnlicgf(uF˜n)da=mdeengt(a∆l

text: gr(otu)p)..

text: BLeytVF˜anbeKoabmtpaienne’ds

text: tfrhoemoreFˆm,byonreepsleaecsinπg

text: DZ,.

text: K

text: Again

text: by

text: Van

text: Kampen’s

text: theorem

text: we

text: see

text: the

text: double

text: of

text: F˜

text: has

text: complement

text: with

text: cyclic

text: fundamental

text: group,

text: and

text: hence

text: by

text: Theorem

text: we

text: know

text: it

text: bounds

text: a

text: handlebody.

text: Therefore,

text: gs

text: (K)

text: deg(∆

text: (t)).

text: top

text: K

text: As

text: we

text: mentioned

text: in

text: the

text: introduction,

text: this

text: theorem

text: has

text: the

text: following

text: immediate

text: corollaries.

text: The

text: ﬁrst

text: one

text: is

text: Corollary

text: which

text: says

text: deg(∆

text: (t))

text: implies

text: K

text: gs

text: (K)

text: top


text: A LOWER BOUND FOR THE DOUBLE SLICE GENUS 13 the canonical basis of ⊕ Z if necessary, so that in terms of these two bases, s can n 9 be represented as a matrixof the form below    1 . . . 0 am+1 . . . an 1 1 ... ... ... ... ... ... 0 . . . 1 am+1 . . . an m m Here the entries of the matrix is counted mod 9.   ⊕ Z Proof. Start with an arbitrary basis for m 9. Write s as a matrix   a1 . . . am am+1 . . . an 1 1 1 1 ... ... ... ... ... ... a1 . . . am am+1 . . . an m m m m ⊕ Z Z Since s is surjective, then the ﬁrst row represents a surjective map to . (cid:45) n 9 9 Therefore, there must exists ai for some i such that 3 ai . So ai is a generator in Z 1 ⊕ Z 1 1 , and we may alter the ﬁrst basis element of by a proper multiplication, 9 m 9 so that under the new basis, ai becomes 1. Reorder the basis for H (# L(9, 4)) if 1 1 n necessary to achieve a1 = 1. 1 Then under the new bases, s has the following matrix form (by abusing notation we still denote the undetermined values by ai):  j    1 . . . am am+1 . . . an 1 1 1 ... ... ... ... ... ... a1 . . . am am+1 . . . an m m m m ⊕ Z Write for now the basis for as e , ..., e . We do a change of basis by letting (cid:48) (cid:48) m 9 1 m { (cid:48)} e = e + Σm a1e , e = e for i = 2, 3, ..., m. Then under the new basis e , 1 1 j=2 j j i i i (abusing notation again) s has matrix form:     1 a2 . . . am am+1 . . . an   1 1 1 1  0 a2 . . . am am+1 . . . an 2 2 m m ... ... ... ... ... ... ... 0 a2 . . . am am+1 . . . an m m m m − Repeat this process at most m 1 times, one for each row, we will achieve a (cid:3) matrix as stated in the lemma. We are ready to prove Proposition 2.10.   Proof of Proposition 2.10. By a properly choice of bases as in Lemma A.1, we may assume s is of the form     1 . . . 0 am+1 . . . an 1 1 ... ... ... ... ... ... 0 . . . 1 am+1 . . . an m m We want to choose a map j : ⊕ Z(cid:2) → Z so that t(cid:3)he corresponding Casson-Gordon m m 9 invariant is big enough. Write j as j j . . . j 1 2 m

text: A

text: LOWER

text: BOUND

text: FOR

text: THE

text: DOUBLE

text: SLICE

text: GENUS

text: the

text: canonical

text: basis

text: of

text: Z

text: if

text: necessary,

text: so

text: that

text: in

text: terms

text: of

text: these

text: two

text: bases,

text: s

text: can

text: n

text: be

text: represented

text: as

text: a

text: matrixof

text: the

text: form

text: below

text: an

text: an

text: m

text: m

text: Here

text: the

text: entries

text: of

text: the

text: matrix

text: is

text: counted

text: mod

text: Z

text: Proof.

text: Start

text: with

text: an

text: arbitrary

text: basis

text: for

text: m

text: Write

text: s

text: as

text: a

text: matrix

text: am

text: an

text: am

text: an

text: m

text: m

text: m

text: m

text: Z

text: Z

text: Since

text: s

text: is

text: surjective,

text: then

text: the

text: ﬁrst

text: row

text: represents

text: a

text: surjective

text: map

text: to

text: n

text: Therefore,

text: there

text: must

text: exists

text: ai

text: for

text: some

text: i

text: such

text: that

text: ai

text: So

text: ai

text: is

text: a

text: generator

text: in

text: Z

text: Z

text: and

text: we

text: may

text: alter

text: the

text: ﬁrst

text: basis

text: element

text: of

text: by

text: a

text: proper

text: multiplication,

text: m

text: so

text: that

text: under

text: the

text: new

text: basis,

text: ai

text: becomes

text: Reorder

text: the

text: basis

text: for

text: H

text: if

text: n

text: necessary

text: to

text: achieve

text: Then

text: under

text: the

text: new

text: bases,

text: s

text: has

text: the

text: following

text: matrix

text: form

text: (by

text: abusing

text: notation

text: we

text: still

text: denote

text: the

text: undetermined

text: values

text: by

text: ai):

text: j

text: am

text: an

text: am

text: an

text: m

text: m

text: m

text: m

text: Z

text: Write

text: for

text: now

text: the

text: basis

text: for

text: as

text: e

text: e

text: We

text: do

text: a

text: change

text: of

text: basis

text: by

text: letting

text: m

text: m

text: e

text: e

text: Σm

text: e

text: e

text: for

text: i

text: m.

text: Then

text: under

text: the

text: new

text: basis

text: e

text: j

text: j

text: i

text: i

text: i

text: (abusing

text: notation

text: again)

text: s

text: has

text: matrix

text: form:

text: 1

text: am

text: an

text: am

text: an

text: m

text: m

text: am

text: an

text: m

text: m

text: m

text: m

text: Repeat

text: this

text: process

text: at

text: most

text: m

text: times,

text: one

text: for

text: each

text: row,

text: we

text: will

text: achieve

text: a

text: matrix

text: as

text: stated

text: in

text: the

text: lemma.

text: We

text: are

text: ready

text: to

text: prove

text: Proposition

text: Proof

text: of

text: Proposition

text: By

text: a

text: properly

text: choice

text: of

text: bases

text: as

text: in

text: Lemma

text: we

text: may

text: assume

text: s

text: is

text: of

text: the

text: form

text: an

text: an

text: m

text: m

text: We

text: want

text: to

text: choose

text: a

text: map

text: j

text: Z

text: so

text: that

text: corresponding

text: Casson-Gordon

text: m

text: m

text: invariant

text: is

text: big

text: enough.

text: Write

text: j

text: as

text: j

text: j

text: j

text: m


text: A LOWER BOUND FOR THE DOUBLE SLICE GENUS 9 The proof of this proposition appears in Appendix A. Clearly Theorem 1.2 will follow from the next theorem. ≥ ≥ Theorem 2.11. θ(K ) n, and hence g (K ) n. 110n ds 110n (cid:122) ⊕(cid:125)(cid:124) (cid:123) Proof. Let (G , G ) be an extendable pair for H (Σ(K )). Since G G is a quo- 1 2 1 110n 1 2 l ⊕ Z ⊕ Z ⊕ · · · ⊕ Z ⊕Z ⊕ tient group of H (Σ(K )) = , we may write G G = · · · ⊕ Z 1 110n 110n 9 ≥ 1 2 9 9 3 for some l. We will prove θ(K ) n by considering the possible values 3 110n of l in two cases. − ≥ ≥ First, if 110n l 2n, then we must have θ (H (Σ), G , G ) n in view 1 1 1 2 of Condition 2 of part (a)-(ii) of Deﬁnition 2.2, since one must employ at least − another 110n l elements of order 9 to get to H (Σ). − 1 Second, if 110n l < 2n, then l > 108n. Then at least one of G , say G , (cid:48) Z (cid:48) ≥ i 1 has l many summands with l 54n. Then by Proposition 2.10 we may ﬁnd 9 → Z → a character φ : H (Σ(K )) that factors through ι : H (Σ(K )) 1 1 110n ≥ 9(cid:48) 1≤ (cid:48) 1 11−0n G , such that σ(Σ(K ), φ ) 10l . Also note that ξ (G ) l + 110n l and 1 110n 1 9 3 1 σ(K ) = 0. Therefore, 110n ≥1 10 (cid:48) − (cid:48) − θ (K , G , G ) ( l (l + 110n l)) 3 110n 1 2 2 9 (cid:48) ≥1 l − ( 2n) 2 9 ≥1 54n − ( 2n) 2 9 =n ≥ (cid:3) Therefore, θ(K) n in either case. Remark. For any given nonnegative integer m, one can similarly prove there is a family of knots whose slice genera are all equal to m and double slice genera grow arbitrarily large. In fact, taking connected sum of m copies of the trefoil knot and K as above would provide such examples. n However, the author is not able to prove the embedding numbers of Σ(K ) grow n arbitrarily large by the Θ-invariant deﬁned in Subsection 2.3. Techniques from [1] give lower bounds for the 2-nd betti number of spin 4-manifolds with boundary being a given 3-manifold, and hence cannot be applied here since our 3-manifolds bound rational homology balls. It seems natural to ask Question 1. Can one ﬁnd a family of rational homology spheres that are bound- aries of spin rational homology balls and whose embedding numbers can grow ar- bitrarily large? Moreover, the author wonder if one can ﬁnd examples so that the Θ-invariant can be applied to answer the above question. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: LOWER

text: BOUND

text: FOR

text: THE

text: DOUBLE

text: SLICE

text: GENUS

text: The

text: proof

text: of

text: this

text: proposition

text: appears

text: in

text: Appendix

text: A.

text: Clearly

text: Theorem

text: will

text: follow

text: from

text: the

text: next

text: theorem.

text: Theorem

text: θ(K

text: n,

text: and

text: hence

text: g

text: (K

text: n.

text: ds

text: Proof.

text: Let

text: (G

text: G

text: be

text: an

text: extendable

text: pair

text: for

text: H

text: (Σ(K

text: Since

text: G

text: G

text: is

text: a

text: quo-

text: l

text: Z

text: Z

text: Z

text: ⊕Z

text: tient

text: group

text: of

text: H

text: (Σ(K

text: we

text: may

text: write

text: G

text: G

text: Z

text: for

text: some

text: l.

text: We

text: will

text: prove

text: θ(K

text: n

text: by

text: considering

text: the

text: possible

text: values

text: of

text: l

text: in

text: two

text: cases.

text: First,

text: if

text: l

text: then

text: we

text: must

text: have

text: θ

text: (H

text: (Σ),

text: G

text: G

text: n

text: in

text: view

text: of

text: Condition

text: of

text: part

text: (a)-(ii)

text: of

text: Deﬁnition

text: since

text: one

text: must

text: employ

text: at

text: least

text: another

text: l

text: elements

text: of

text: order

text: to

text: get

text: to

text: H

text: (Σ).

text: Second,

text: if

text: l

text: then

text: l

text: Then

text: at

text: least

text: one

text: of

text: G

text: say

text: G

text: Z

text: i

text: has

text: l

text: many

text: summands

text: with

text: l

text: Then

text: by

text: Proposition

text: we

text: may

text: ﬁnd

text: Z

text: a

text: character

text: φ

text: H

text: (Σ(K

text: that

text: factors

text: through

text: ι

text: H

text: (Σ(K

text: 1≤

text: G

text: such

text: that

text: σ(Σ(K

text: φ

text: Also

text: note

text: that

text: ξ

text: (G

text: l

text: l

text: and

text: σ(K

text: Therefore,

text: ≥1

text: θ

text: (K

text: G

text: G

text: l

text: (l

text: l))

text: ≥1

text: l

text: ≥1

text: =n

text: Therefore,

text: θ(K)

text: n

text: in

text: either

text: case.

text: Remark.

text: For

text: any

text: given

text: nonnegative

text: integer

text: m,

text: one

text: can

text: similarly

text: prove

text: there

text: is

text: a

text: family

text: of

text: knots

text: whose

text: slice

text: genera

text: are

text: all

text: equal

text: to

text: m

text: and

text: double

text: slice

text: genera

text: grow

text: arbitrarily

text: large.

text: In

text: fact,

text: taking

text: connected

text: sum

text: of

text: m

text: copies

text: of

text: the

text: trefoil

text: knot

text: and

text: K

text: as

text: above

text: would

text: provide

text: such

text: examples.

text: n

text: However,

text: the

text: author

text: is

text: not

text: able

text: to

text: prove

text: the

text: embedding

text: numbers

text: of

text: Σ(K

text: grow

text: n

text: arbitrarily

text: large

text: by

text: the

text: Θ-invariant

text: deﬁned

text: in

text: Subsection

text: Techniques

text: from

text: give

text: lower

text: bounds

text: for

text: the

text: betti

text: number

text: of

text: spin

text: with

text: boundary

text: being

text: a

text: given

text: and

text: hence

text: cannot

text: be

text: applied

text: here

text: since

text: our

text: bound

text: rational

text: homology

text: balls.

text: It

text: seems

text: natural

text: to

text: ask

text: Question

text: Can

text: one

text: ﬁnd

text: a

text: family

text: of

text: rational

text: homology

text: spheres

text: that

text: are

text: bound-

text: aries

text: of

text: spin

text: rational

text: homology

text: balls

text: and

text: whose

text: embedding

text: numbers

text: can

text: grow

text: ar-

text: bitrarily

text: large?

text: Moreover,

text: the

text: author

text: wonder

text: if

text: one

text: can

text: ﬁnd

text: examples

text: so

text: that

text: the

text: Θ-invariant

text: can

text: be

text: applied

text: to

text: answer

text: the

text: above

text: question.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: HAUSDORFF DIMENSION OF KUPERBERG MINIMAL SETS 3 A vector ﬁeld on a manifold is said to have a closed orbit if one of its integral curves is homeomorphic to S1. The Hopf vector ﬁeld on S3, whose integral curves form the Hopf ﬁbration, has all orbits closed. In 1950, Seifert ([44]) showed that every nonsingular vector ﬁeld on S3 suﬃciently close to the Hopf vector ﬁeld also has a closed orbit, and then asked if every continuous vector ﬁeld on S3 does. The generalized Seifert conjecture asked this question for any compact n-manifold with Euler characteristic zero. Counterexamples in dimension four and greater were discovered in 1966 by Wilson ([55]), who constructed the ﬁrst plug, the product of a closed rectangle with a torus, which carries a smooth vector ﬁeld satisfying certain properties. A plug is a manifold with boundary, together with a smooth vector ﬁeld. If this local vector ﬁeld satisﬁes some symmetry conditions, the plug can be inserted into a manifold carrying a global vector ﬁeld, in such a way that the local dynamics in the plug are compatible with the global dynamics. If the plug intersects a periodic orbit, the plug’s interior dynamics can break it. Using this method, Wilson constructed smooth counterexamples to Seifert’s conjecture in dimension greater than or equal to four. Seifert’s conjecture is vacuous in dimension two, so Seifert’s conjecture only remained in dimension three, although Wilson did succeed in showing that on every closed connected three-manifold, there exists a smooth vector ﬁeld with only ﬁnitely many closed orbits. The ﬁrst counterexample to Seifert’s conjecture in dimension three was constructed in 1972 by Schweitzer ([43]). This counterexample used a plug supporting an aperiodic vector ﬁeld of class C1. In 1988, Harrison ([18]) modiﬁed Schweitzer’s construction to class C2, but serious obstructions remained in extending to C∞. For an account of Schweitzer’s and Harrison’s constructions, see [15]. In 1994, Kuperberg ([26]) constructed a C∞ counterexample to Seifert’s conjecture in R dimension three. This construction used a modiﬁed Wilson plug embedded in 3 with self- intersections. Kuperberg used these self-intersections to break the periodic orbits inside Wilson’s plug without creating new periodic orbits. See [26], [15], [20], [30] for descriptions of Kuperberg’s construction. Ghys ([15]) showed that Kuperberg’s plug contains a unique minimal set. Using a numerical simulation due to B. Sevannec, he obtained an image of this minimal set on a transverse section of the plug. See Figure 1. Ghys encouraged an investigation into the properties of this minimal set, and asked how such properties depend on the Kuperberg’s construction. A closer study of the topology and dynamics of the minimal set was carried out by Hurder and Rechtman ([20]). To answer Ghys’ question, they formulated a special class of ﬂows called generic Kuperberg ﬂows that preserves a unique minimal set with the following characterization. Theorem 1.1. ([20], Theorem 17.1) Let M ⊂ K be the minimal set of a generic Kuperberg ﬂow. Then M is a codimension one lamination with radial Cantor transversal M ∩ [0, 1].

text: HAUSDORFF

text: DIMENSION

text: OF

text: KUPERBERG

text: MINIMAL

text: SETS

text: A

text: vector

text: ﬁeld

text: on

text: a

text: manifold

text: is

text: said

text: to

text: have

text: a

text: closed

text: orbit

text: if

text: one

text: of

text: its

text: integral

text: curves

text: is

text: homeomorphic

text: to

text: The

text: Hopf

text: vector

text: ﬁeld

text: on

text: whose

text: integral

text: curves

text: form

text: the

text: Hopf

text: ﬁbration,

text: has

text: all

text: orbits

text: closed.

text: In

text: Seifert

text: showed

text: that

text: every

text: nonsingular

text: vector

text: ﬁeld

text: on

text: suﬃciently

text: close

text: to

text: the

text: Hopf

text: vector

text: ﬁeld

text: also

text: has

text: a

text: closed

text: orbit,

text: and

text: then

text: asked

text: if

text: every

text: continuous

text: vector

text: ﬁeld

text: on

text: does.

text: The

text: generalized

text: Seifert

text: conjecture

text: asked

text: this

text: question

text: for

text: any

text: compact

text: n-manifold

text: with

text: Euler

text: characteristic

text: zero.

text: Counterexamples

text: in

text: dimension

text: four

text: and

text: greater

text: were

text: discovered

text: in

text: by

text: Wilson

text: who

text: constructed

text: the

text: ﬁrst

text: plug,

text: the

text: product

text: of

text: a

text: closed

text: rectangle

text: with

text: a

text: torus,

text: which

text: carries

text: a

text: smooth

text: vector

text: ﬁeld

text: satisfying

text: certain

text: properties.

text: A

text: plug

text: is

text: a

text: manifold

text: with

text: boundary,

text: together

text: with

text: a

text: smooth

text: vector

text: ﬁeld.

text: If

text: this

text: local

text: vector

text: ﬁeld

text: satisﬁes

text: some

text: symmetry

text: conditions,

text: the

text: plug

text: can

text: be

text: inserted

text: into

text: a

text: manifold

text: carrying

text: a

text: global

text: vector

text: ﬁeld,

text: in

text: such

text: a

text: way

text: that

text: the

text: local

text: dynamics

text: in

text: the

text: plug

text: are

text: compatible

text: with

text: the

text: global

text: dynamics.

text: If

text: the

text: plug

text: intersects

text: a

text: periodic

text: orbit,

text: the

text: plug’s

text: interior

text: dynamics

text: can

text: break

text: it.

text: Using

text: this

text: method,

text: Wilson

text: constructed

text: smooth

text: counterexamples

text: to

text: Seifert’s

text: conjecture

text: in

text: dimension

text: greater

text: than

text: or

text: equal

text: to

text: four.

text: Seifert’s

text: conjecture

text: is

text: vacuous

text: in

text: dimension

text: two,

text: so

text: Seifert’s

text: conjecture

text: only

text: remained

text: in

text: dimension

text: three,

text: although

text: Wilson

text: did

text: succeed

text: in

text: showing

text: that

text: on

text: every

text: closed

text: connected

text: three-manifold,

text: there

text: exists

text: a

text: smooth

text: vector

text: ﬁeld

text: with

text: only

text: ﬁnitely

text: many

text: closed

text: orbits.

text: The

text: ﬁrst

text: counterexample

text: to

text: Seifert’s

text: conjecture

text: in

text: dimension

text: three

text: was

text: constructed

text: in

text: by

text: Schweitzer

text: This

text: counterexample

text: used

text: a

text: plug

text: supporting

text: an

text: aperiodic

text: vector

text: ﬁeld

text: of

text: class

text: In

text: Harrison

text: modiﬁed

text: Schweitzer’s

text: construction

text: to

text: class

text: but

text: serious

text: obstructions

text: remained

text: in

text: extending

text: to

text: C∞.

text: For

text: an

text: account

text: of

text: Schweitzer’s

text: and

text: Harrison’s

text: constructions,

text: see

text: In

text: Kuperberg

text: constructed

text: a

text: C∞

text: counterexample

text: to

text: Seifert’s

text: conjecture

text: in

text: R

text: dimension

text: three.

text: This

text: construction

text: used

text: a

text: modiﬁed

text: Wilson

text: plug

text: embedded

text: in

text: with

text: self-

text: intersections.

text: Kuperberg

text: used

text: these

text: self-intersections

text: to

text: break

text: the

text: periodic

text: orbits

text: inside

text: Wilson’s

text: plug

text: without

text: creating

text: new

text: periodic

text: orbits.

text: See

text: for

text: descriptions

text: of

text: Kuperberg’s

text: construction.

text: Ghys

text: showed

text: that

text: Kuperberg’s

text: plug

text: contains

text: a

text: unique

text: minimal

text: set.

text: Using

text: a

text: numerical

text: simulation

text: due

text: to

text: B.

text: Sevannec,

text: he

text: obtained

text: an

text: image

text: of

text: this

text: minimal

text: set

text: on

text: a

text: transverse

text: section

text: of

text: the

text: plug.

text: See

text: Figure

text: Ghys

text: encouraged

text: an

text: investigation

text: into

text: the

text: properties

text: of

text: this

text: minimal

text: set,

text: and

text: asked

text: how

text: such

text: properties

text: depend

text: on

text: the

text: Kuperberg’s

text: construction.

text: A

text: closer

text: study

text: of

text: the

text: topology

text: and

text: dynamics

text: of

text: the

text: minimal

text: set

text: was

text: carried

text: out

text: by

text: Hurder

text: and

text: Rechtman

text: To

text: answer

text: Ghys’

text: question,

text: they

text: formulated

text: a

text: special

text: class

text: of

text: ﬂows

text: called

text: generic

text: Kuperberg

text: ﬂows

text: that

text: preserves

text: a

text: unique

text: minimal

text: set

text: with

text: the

text: following

text: characterization.

text: Theorem

text: Theorem

text: Let

text: M

text: K

text: be

text: the

text: minimal

text: set

text: of

text: a

text: generic

text: Kuperberg

text: ﬂow.

text: Then

text: M

text: is

text: a

text: codimension

text: one

text: lamination

text: with

text: radial

text: Cantor

text: transversal

text: M


text: 22 DANIEL INGEBRETSON By the nesting property, for any n ≥ 1 and ω ∈ Σ we have a map φ : X → X given by ω the composition φ = φ ◦ φ ◦ · · · ◦ φ ◦ f . ω ω1,ω2 ω2,ω3 ωn−1,ωn ωn For each ω ∈ Σ deﬁne n ∆ = φ (X). ω ω As a consequence of the nesting property, we have ∆ ⊂ ∆ , and ∆ ∩ ∆ (cid:54)= ∅ ω,i ω ω,i ω,j for all ω ∈ Σ ∩ E∗ and i (cid:54)= j ∈ E such that (ω, i) and (ω, j) ∈ Σ ∩ E∗. Because the maps φ have global L(cid:0) ipsch(cid:1)itz constant 0 < s < 1, we have for each n ≥ 1 i,j that (cid:84) diam ∆ ≤ sndiam(X). ω|n ∞ As with the graph-directed systems, the compact sets ∆ are nested, so ∆ is ω|n n=1 ω|n necessarily a singleton and nonempty by our assumption on Σ. This deﬁnes a coding map (cid:92) π : Σ → X given by ∞ π(ω) = ∆ . ω|n n=1 The limit set J of the general function system {φ } is i,j (cid:91) (cid:92) (5) J = π(Σ) ∞ = (cid:92) (cid:91) ∆ ω|n ω∈Σ n=1 ∞ = ∆ . ω n=1 ω∈Σn As with graph-directed systems, for our applications we will only consider the case when X ⊂ [0, 1] is compact and each ∆ ⊂ X is a closed subinterval. i We will impose the same regularity conditions on general function systems as we did on graph-directed systems. Namely, we assume that each map f : X → X and φ : ∆ → X i i,j j has regularity C1+α for some α > 0. We call such a function system a C1+α general function system modeled by Σ. If A : E × E → {0, 1} is an incidence matrix deﬁning admissible words En as in Section A , the choices Σ∗ = E∗ , Σ∞ = E∞ and Σ = Σ∗ ∪ Σ∞ makes the general function system a A A graph directed pseudo-Markov system as in Section 3.1, so graph directed pseudo-Markov systems are a special case of general function systems.

text: DANIEL

text: INGEBRETSON

text: By

text: the

text: nesting

text: property,

text: for

text: any

text: n

text: and

text: ω

text: Σ

text: we

text: have

text: a

text: map

text: φ

text: X

text: X

text: given

text: by

text: ω

text: the

text: composition

text: φ

text: φ

text: φ

text: φ

text: f

text: ω

text: ωn

text: For

text: each

text: ω

text: Σ

text: deﬁne

text: n

text: φ

text: (X).

text: ω

text: ω

text: As

text: a

text: consequence

text: of

text: the

text: nesting

text: property,

text: we

text: have

text: and

text: ω,i

text: ω

text: ω,i

text: ω,j

text: for

text: all

text: ω

text: Σ

text: E∗

text: and

text: i

text: j

text: E

text: such

text: that

text: (ω,

text: i)

text: and

text: (ω,

text: j)

text: Σ

text: E∗.

text: Because

text: the

text: maps

text: φ

text: have

text: global

text: constant

text: s

text: we

text: have

text: for

text: each

text: n

text: i,j

text: that

text: diam

text: sndiam(X).

text: ω|n

text: As

text: with

text: the

text: graph-directed

text: systems,

text: the

text: compact

text: sets

text: are

text: nested,

text: so

text: is

text: ω|n

text: ω|n

text: necessarily

text: a

text: singleton

text: and

text: nonempty

text: by

text: our

text: assumption

text: on

text: Σ.

text: This

text: deﬁnes

text: a

text: coding

text: map

text: π

text: Σ

text: X

text: given

text: by

text: π(ω)

text: ω|n

text: The

text: limit

text: set

text: J

text: of

text: the

text: general

text: function

text: system

text: {φ

text: is

text: i,j

text: J

text: π(Σ)

text: ω|n

text: ω∈Σ

text: ω

text: ω∈Σn

text: As

text: with

text: graph-directed

text: systems,

text: for

text: our

text: applications

text: we

text: will

text: only

text: consider

text: the

text: case

text: when

text: X

text: is

text: compact

text: and

text: each

text: X

text: is

text: a

text: closed

text: subinterval.

text: i

text: We

text: will

text: impose

text: the

text: same

text: regularity

text: conditions

text: on

text: general

text: function

text: systems

text: as

text: we

text: did

text: on

text: graph-directed

text: systems.

text: Namely,

text: we

text: assume

text: that

text: each

text: map

text: f

text: X

text: X

text: and

text: φ

text: X

text: i

text: i,j

text: j

text: has

text: regularity

text: for

text: some

text: α

text: We

text: call

text: such

text: a

text: function

text: system

text: a

text: general

text: function

text: system

text: modeled

text: by

text: Σ.

text: If

text: A

text: E

text: E

text: is

text: an

text: incidence

text: matrix

text: deﬁning

text: admissible

text: words

text: En

text: as

text: in

text: Section

text: A

text: the

text: choices

text: Σ∗

text: E∗

text: Σ∞

text: E∞

text: and

text: Σ

text: Σ∗

text: Σ∞

text: makes

text: the

text: general

text: function

text: system

text: a

text: A

text: A

text: graph

text: directed

text: pseudo-Markov

text: system

text: as

text: in

text: Section

text: so

text: graph

text: directed

text: pseudo-Markov

text: systems

text: are

text: a

text: special

text: case

text: of

text: general

text: function

text: systems.


text: 26  DANIEL INGEBRETSON   AE(i, j) : i, j ∈ E (6) AE∪F (i, j) = AF (i, j) : i, j ∈ F 1 : i ∈ E, j ∈ F or i ∈ F, j ∈ E Consider the intervals X = φ (X) for ω ∈ En and Y = ψ (X) for τ ∈ F n . Then if ω (cid:92) ω(cid:91) AE (cid:92)τ (cid:91) τ AF J and J are the limit sets of the respective pseudo-Markov systems, we have by Equation E F 5 that ∞ ∞ J = X , and J = Y . E ω F ω n=1 ω∈En n=1 τ∈Fn AE AF The interlacing J of J and J is th(cid:92)e limit s(cid:91)et of the joint pseudo-Markov system, and E∪F E F is given by ∞ J = ∆ , E∪F ω n=1 ω∈(E∪F)n AE∪F where ∆ = γ (X) and γ is the composition of the maps φ and ψ indexed by admissible ω ω ω i,j i,j words ω in the joint sequence space (E ∪ F )n . AE∪f

label: equation

text: DANIEL

text: INGEBRETSON

text: AE(i,

text: j)

text: i,

text: j

text: E

text: AE∪F

text: (i,

text: j)

text: AF

text: (i,

text: j)

text: i,

text: j

text: F

text: i

text: E,

text: j

text: F

text: or

text: i

text: F,

text: j

text: E

text: Consider

text: the

text: intervals

text: X

text: φ

text: (X)

text: for

text: ω

text: En

text: and

text: Y

text: ψ

text: (X)

text: for

text: τ

text: F

text: n

text: Then

text: if

text: ω

text: AE

text: τ

text: AF

text: J

text: and

text: J

text: are

text: the

text: limit

text: sets

text: of

text: the

text: respective

text: pseudo-Markov

text: systems,

text: we

text: have

text: by

text: Equation

text: E

text: F

text: that

text: J

text: X

text: and

text: J

text: Y

text: E

text: ω

text: F

text: ω

text: ω∈En

text: τ∈Fn

text: AE

text: AF

text: The

text: interlacing

text: J

text: of

text: J

text: and

text: J

text: is

text: limit

text: of

text: the

text: joint

text: pseudo-Markov

text: system,

text: and

text: E∪F

text: E

text: F

text: is

text: given

text: by

text: J

text: E∪F

text: ω

text: ω∈(E∪F)n

text: AE∪F

text: where

text: γ

text: (X)

text: and

text: γ

text: is

text: the

text: composition

text: of

text: the

text: maps

text: φ

text: and

text: ψ

text: indexed

text: by

text: admissible

text: ω

text: ω

text: ω

text: i,j

text: i,j

text: words

text: ω

text: in

text: the

text: joint

text: sequence

text: space

text: (E

text: F

text: )n

text: AE∪f


text: 28 DANIEL INGEBRETSON (cid:80) A generalization of Bowen’s equation ([7]) is proved in [51] for what are termed “weakly thin” pseudo-Markov systems. Weak thinness is a general notion, but in our setting it is equivalent to p (1) = |∆ |< ∞, which is a consequence of our separation condition in 1 i∈E i Deﬁnition 3.1. Theorem 4.2 (Proposition 4.13 of [51]). Let {φ } be a C1+α pseudo-Markov system with i,j limit set J and associated pressure function p(t). Then the Hausdorﬀ dimension dim (J) H satisﬁes dim (J) = inf{t ≥ 0 : p(t) < 0}, H and if p(t) = 0 then t is the only zero of p(t) and t = dim (J). H In Section we introduced the asymptotically stationary pseudo-Markov systems, with error a±. We assume that this error is summable and monotone, as speciﬁed in that section. The δ dimension theory of stationary systems is particularly simple and goes back to Moran ([34]). The dimension theory of asymptotically stationary systems is similar. Theorem 4.3. Let {φ } be an asymptotically stationary pseudo-Markov system, with sum- i,j mable monotone error a±, and let J be its limit set. Then the Lebesgue measure of J δ δ δ satisﬁes lim µ(J ) = 0, δ δ→0 and the Hausdorﬀ dimension dim (J ) satisﬁes H δ 0 < lim dim (J ) < 1. H δ δ→0     (cid:92) (cid:91) (cid:91) (cid:88) Proof. Let µ be Lebesgue measure on [0, 1]. By the nesting and separation conditions on ∆ ,     ω ∞ µ(J ) = µ ∆ = lim µ ∆ = lim |∆ |. δ ω ω ω n→∞ n→∞ n=1 ω∈En(cid:88) ω∈En (cid:88) ω∈En A A A We then substitute Equation 4 to obtain µ(J ) ≤ lim (cid:88) s r · · · r + lim (cid:88) a+(ω) δ n→∞ ω1 ω2 ωn n→∞ δ ω∈En ω∈En A A ≤ lim (cid:32) s (cid:33)r(cid:32) · · · r (cid:33)+ lim a+(ω) n→∞ (cid:88) ω1 ω2(cid:88) ωn n→∞ (cid:88)δ ω∈En ω∈En A n−1 = lim s r + lim a+(ω) n→∞ i i n→∞ δ i∈E i∈E ω∈En A

text: DANIEL

text: INGEBRETSON

text: A

text: generalization

text: of

text: Bowen’s

text: equation

text: is

text: proved

text: in

text: for

text: what

text: are

text: termed

text: “weakly

text: thin”

text: pseudo-Markov

text: systems.

text: Weak

text: thinness

text: is

text: a

text: general

text: notion,

text: but

text: in

text: our

text: setting

text: it

text: is

text: equivalent

text: to

text: p

text: which

text: is

text: a

text: consequence

text: of

text: our

text: separation

text: condition

text: in

text: i∈E

text: i

text: Deﬁnition

text: Theorem

text: (Proposition

text: of

text: Let

text: {φ

text: be

text: a

text: pseudo-Markov

text: system

text: with

text: i,j

text: limit

text: set

text: J

text: and

text: associated

text: pressure

text: function

text: p(t).

text: Then

text: the

text: Hausdorﬀ

text: dimension

text: dim

text: (J)

text: H

text: satisﬁes

text: dim

text: (J)

text: inf{t

text: p(t)

text: H

text: and

text: if

text: p(t)

text: then

text: t

text: is

text: the

text: only

text: zero

text: of

text: p(t)

text: and

text: t

text: dim

text: (J).

text: H

text: In

text: Section

text: we

text: introduced

text: the

text: asymptotically

text: stationary

text: pseudo-Markov

text: systems,

text: with

text: error

text: a±.

text: We

text: assume

text: that

text: this

text: error

text: is

text: summable

text: and

text: monotone,

text: as

text: speciﬁed

text: in

text: that

text: section.

text: The

text: δ

text: dimension

text: theory

text: of

text: stationary

text: systems

text: is

text: particularly

text: simple

text: and

text: goes

text: back

text: to

text: Moran

text: The

text: dimension

text: theory

text: of

text: asymptotically

text: stationary

text: systems

text: is

text: similar.

text: Theorem

text: Let

text: {φ

text: be

text: an

text: asymptotically

text: stationary

text: pseudo-Markov

text: system,

text: with

text: sum-

text: i,j

text: mable

text: monotone

text: error

text: a±,

text: and

text: let

text: J

text: be

text: its

text: limit

text: set.

text: Then

text: the

text: Lebesgue

text: measure

text: of

text: J

text: δ

text: δ

text: δ

text: satisﬁes

text: lim

text: µ(J

text: δ

text: δ→0

text: and

text: the

text: Hausdorﬀ

text: dimension

text: dim

text: (J

text: satisﬁes

text: H

text: δ

text: lim

text: dim

text: (J

text: H

text: δ

text: δ→0

text: Proof.

text: Let

text: µ

text: be

text: Lebesgue

text: measure

text: on

text: By

text: the

text: nesting

text: and

text: separation

text: conditions

text: on

text: ω

text: µ(J

text: µ

text: lim

text: µ

text: lim

text: δ

text: ω

text: ω

text: ω

text: n→∞

text: n→∞

text: ω∈En

text: ω∈En

text: A

text: A

text: A

text: We

text: then

text: substitute

text: Equation

text: to

text: obtain

text: µ(J

text: lim

text: s

text: r

text: r

text: lim

text: a+(ω)

text: δ

text: n→∞

text: ω1

text: ω2

text: ωn

text: n→∞

text: δ

text: ω∈En

text: ω∈En

text: A

text: A

text: lim

text: s

text: r

text: lim

text: a+(ω)

text: n→∞

text: ω1

text: ωn

text: n→∞

text: ω∈En

text: ω∈En

text: A

text: lim

text: s

text: r

text: lim

text: a+(ω)

text: n→∞

text: i

text: i

text: n→∞

text: δ

text: i∈E

text: i∈E

text: ω∈En

text: A


text: (cid:80) HAUSDORFF DIMENSION OF KUPERBERG MINIMAL SETS 29 By the separation condition in Deﬁnition 3.1 we know r < 1. By the summability and i∈E i monotonicity conditions on a+ the right term decreases to 0 as δ → 0, so lim µ(J ) = 0, δ δ→0 δ as desired. We now turn to the Hausdorﬀ dimension. Let p (t) be the pressure function associated to δ this pseudo-Markov system. Substituting Equation 4 into the pressure function in Equation 8 we obtain that p− < p < p+, where (cid:88) (cid:88) δ δ δ   1 p±(t) = lim log (s r · · · r )t ± a±(ω) δ n→∞ n ω1 ω2 ωn δ ω∈En ω∈En A A (cid:88) Then by the monotonicity of a± we have that lim p± = p, where δ δ→0 δ 1 p(t) = lim log (s r · · · r )t n→∞ n ω1 ω2 ωn ω∈En (cid:88) A For the upper bound, we calculate p(t) < lim 1 log (cid:32) st (cid:33)rt(cid:32) · · · rt(cid:33) n→∞ n (cid:88) ω1 ω2(cid:88) ωn ω∈En n−1 (cid:88)1 = lim log st rt n→∞ n i i i∈E i∈E = lo(cid:80)g rt. i i∈E Let t be the unique solution to rt = 1, and notice that t < 1. Applying Bowen’s ∗ i∈E i ∗ theorem (4.2), we have dim (J) < t < 1. H ∗ For the lower bound, recall that for all n ≥ 1, Encontains more thanone word, say A (cid:88) ω = (ω , . . . , ω ). 1 n   n−1 1 1 p(t) > lim log (s r · · · r )t = lim log st + log rt . n→∞ n ω1 ω2 ωn n→∞ n ωn ωj j=1 Setting the right hand side = 0, we see that t = 0 is a solution. So again by Bowen’s ∗ (cid:3) theorem, we have dim (J) > t = 0. H ∗ Recall the following deﬁnition of local Hausdorﬀ dimension (see [24]). Let C ⊂ X be a compact subspace of a metric space, and let B (x) denote a ball of radius r centered at x. r For any x ∈ C, the local Hausdorﬀ dimension dim (x) at x is loc (9) dim (x) = lim dim (B (x)) . loc H (cid:15) (cid:15)→0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: HAUSDORFF

text: DIMENSION

text: OF

text: KUPERBERG

text: MINIMAL

text: SETS

text: By

text: the

text: separation

text: condition

text: in

text: Deﬁnition

text: we

text: know

text: r

text: By

text: the

text: summability

text: and

text: i∈E

text: i

text: monotonicity

text: conditions

text: on

text: a+

text: the

text: right

text: term

text: decreases

text: to

text: as

text: δ

text: so

text: lim

text: µ(J

text: δ

text: δ→0

text: δ

text: as

text: desired.

text: We

text: now

text: turn

text: to

text: the

text: Hausdorﬀ

text: dimension.

text: Let

text: p

text: (t)

text: be

text: the

text: pressure

text: function

text: associated

text: to

text: δ

text: this

text: pseudo-Markov

text: system.

text: Substituting

text: Equation

text: into

text: the

text: pressure

text: function

text: in

text: Equation

text: we

text: obtain

text: that

text: p−

text: p

text: p+,

text: where

text: δ

text: δ

text: δ

text: p±(t)

text: lim

text: log

text: (s

text: r

text: r

text: )t

text: a±(ω)

text: δ

text: n→∞

text: n

text: ω1

text: ω2

text: ωn

text: δ

text: ω∈En

text: ω∈En

text: A

text: A

text: Then

text: by

text: the

text: monotonicity

text: of

text: a±

text: we

text: have

text: that

text: lim

text: p±

text: p,

text: where

text: δ

text: δ→0

text: δ

text: p(t)

text: lim

text: log

text: (s

text: r

text: r

text: )t

text: n→∞

text: n

text: ω1

text: ω2

text: ωn

text: ω∈En

text: A

text: For

text: the

text: upper

text: bound,

text: we

text: calculate

text: p(t)

text: lim

text: log

text: st

text: n→∞

text: n

text: ω1

text: ωn

text: ω∈En

text: lim

text: log

text: st

text: rt

text: n→∞

text: n

text: i

text: i

text: i∈E

text: i∈E

text: rt.

text: i

text: i∈E

text: Let

text: t

text: be

text: the

text: unique

text: solution

text: to

text: rt

text: and

text: notice

text: that

text: t

text: Applying

text: Bowen’s

text: i∈E

text: i

text: theorem

text: we

text: have

text: dim

text: (J)

text: t

text: H

text: For

text: the

text: lower

text: bound,

text: recall

text: that

text: for

text: all

text: n

text: Encontains

text: more

text: thanone

text: word,

text: say

text: A

text: ω

text: (ω

text: ω

text: n

text: p(t)

text: lim

text: log

text: (s

text: r

text: r

text: )t

text: lim

text: log

text: st

text: log

text: rt

text: n→∞

text: n

text: ω1

text: ω2

text: ωn

text: n→∞

text: n

text: ωn

text: ωj

text: Setting

text: the

text: right

text: hand

text: side

text: we

text: see

text: that

text: t

text: is

text: a

text: solution.

text: So

text: again

text: by

text: Bowen’s

text: theorem,

text: we

text: have

text: dim

text: (J)

text: t

text: H

text: Recall

text: the

text: following

text: deﬁnition

text: of

text: local

text: Hausdorﬀ

text: dimension

text: (see

text: Let

text: C

text: X

text: be

text: a

text: compact

text: subspace

text: of

text: a

text: metric

text: space,

text: and

text: let

text: B

text: (x)

text: denote

text: a

text: ball

text: of

text: radius

text: r

text: centered

text: at

text: x.

text: r

text: For

text: any

text: x

text: C,

text: the

text: local

text: Hausdorﬀ

text: dimension

text: dim

text: (x)

text: at

text: x

text: is

text: loc

text: dim

text: (x)

text: lim

text: dim

text: (B

text: (x))

text: loc

text: H

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: HAUSDORFF DIMENSION OF KUPERBERG MINIMAL SETS 49 (cid:91) (28) O+(x) ∩ S = g(x). i g∈Ψi,1 To further simplify the analysis, we will focus on the case of O+(x) ∩ S. To save on 1 notation, we will denote by Ψ, Φ, σ, γ, γ the objects Ψ , Φ , Θ , σ , γ , γ respectively, c 1,1 1,1 1,1 1 1 c,1 as we did in Section . In Section , we introduced the level decomposition of orbits and orbit surfaces. For i = 1 (cid:91) the orbit intersection from Equation 28 has a level decomposition ∞ O+(x) ∩ S = O+(x) ∩ S, where O+(x) ∩ S = {y ∈ O+(x) ∩ S : n (y) = k}. 1 1 k 1 k 1 x k=0 The sub-pseudogroup (cid:104)Φ, Θ(cid:105) permutes this level decomposition in the following way. Lemma 7.2. For any k ≥ 0, the map Φ restricted to O+(x) ∩ S maps O+(x) ∩ S into 1 1 k O+(x) ∩ S. 1 k Proof. Let y ∈ O+(x) ∩ S, so there exists T > 0 such that y = ψ (x) and n (T ) = k. Since 1 k T x S ⊂ D−, y is necessarily a secondary entrance point. By Lemma 6.4, there exists U > T such that ψ (x) ∈ D+ and n (T ) = n (U) = k. If Φ(y) is deﬁned, Φ(y) = φ (y) where T (cid:48) > 0 U x x T(cid:48) is the minimal time such that φ (y) ∈ S. Notice that T (cid:48) > U − T and the orbit segment T(cid:48) O+(y, U − T, T (cid:48)) does not intersect D+– if it did, it would have ﬁrst had to intersect D− and (cid:3) T (cid:48) is the minimal such time that this can occur. Thus n (T (cid:48)) = k and Φ(y) ∈ O+(x) ∩S. x 1 k Lemma 7.3. For any k ≥ 0, the map Θ restricted to O+(x) ∩ S maps O+(x) ∩ S into 1 1 k O+(x) ∩ S. 1 k+1 Proof. Let y ∈ O+(x) ∩ S, so there exists T > 0 such that y = ψ (x) and n (T ) = k. As 1 k T x in Lemma 7.2, y is a secondary entrance point. Let U > T be the minimal time such that ψ (x) ∈ D+ ∪ D− is a secondary transition point. Then for any T < t ≤ U, n (t) = k + 1 U x by deﬁnition of the level function. If Θ(y) is deﬁned, Θ(y) = φ (σ−1y) where T (cid:48) > 0 is the T(cid:48) minimal time such that φ (σ−1y) ∈ S. Such a point is a secondary transition point, so in T(cid:48) (cid:3) particular T (cid:48) ≤ U and n (T (cid:48)) = k + 1; i.e. Θ(y) ∈ O+(x) ∩ S. x 1 k+1 The intersection O+(η) ∩ S of a forward orbit surface has a a similar level decomposition, 1 and (cid:104)Φ, Θ(cid:105) acts this level decomposition in the same way.

text: HAUSDORFF

text: DIMENSION

text: OF

text: KUPERBERG

text: MINIMAL

text: SETS

text: O+(x)

text: S

text: g(x).

text: i

text: To

text: further

text: simplify

text: the

text: analysis,

text: we

text: will

text: focus

text: on

text: the

text: case

text: of

text: O+(x)

text: S.

text: To

text: save

text: on

text: notation,

text: we

text: will

text: denote

text: by

text: Ψ,

text: Φ,

text: σ,

text: γ,

text: γ

text: the

text: objects

text: Ψ

text: Φ

text: Θ

text: σ

text: γ

text: γ

text: respectively,

text: c

text: as

text: we

text: did

text: in

text: Section

text: In

text: Section

text: we

text: introduced

text: the

text: level

text: decomposition

text: of

text: orbits

text: and

text: orbit

text: surfaces.

text: For

text: i

text: the

text: orbit

text: intersection

text: from

text: Equation

text: has

text: a

text: level

text: decomposition

text: O+(x)

text: S

text: O+(x)

text: S,

text: where

text: O+(x)

text: S

text: {y

text: O+(x)

text: S

text: n

text: (y)

text: k}.

text: k

text: k

text: x

text: The

text: sub-pseudogroup

text: permutes

text: this

text: level

text: decomposition

text: in

text: the

text: following

text: way.

text: Lemma

text: For

text: any

text: k

text: the

text: map

text: Φ

text: restricted

text: to

text: O+(x)

text: S

text: maps

text: O+(x)

text: S

text: into

text: k

text: O+(x)

text: S.

text: k

text: Proof.

text: Let

text: y

text: O+(x)

text: S,

text: so

text: there

text: exists

text: T

text: such

text: that

text: y

text: ψ

text: (x)

text: and

text: n

text: (T

text: k.

text: Since

text: k

text: T

text: x

text: S

text: D−,

text: y

text: is

text: necessarily

text: a

text: secondary

text: entrance

text: point.

text: By

text: Lemma

text: there

text: exists

text: U

text: T

text: such

text: that

text: ψ

text: (x)

text: D+

text: and

text: n

text: (T

text: n

text: (U)

text: k.

text: If

text: Φ(y)

text: is

text: deﬁned,

text: Φ(y)

text: φ

text: (y)

text: where

text: T

text: U

text: x

text: x

text: is

text: the

text: minimal

text: time

text: such

text: that

text: φ

text: (y)

text: S.

text: Notice

text: that

text: T

text: U

text: T

text: and

text: the

text: orbit

text: segment

text: O+(y,

text: U

text: T,

text: T

text: does

text: not

text: intersect

text: D+–

text: if

text: it

text: did,

text: it

text: would

text: have

text: ﬁrst

text: had

text: to

text: intersect

text: D−

text: and

text: T

text: is

text: the

text: minimal

text: such

text: time

text: that

text: this

text: can

text: occur.

text: Thus

text: n

text: (T

text: k

text: and

text: Φ(y)

text: O+(x)

text: ∩S.

text: x

text: k

text: Lemma

text: For

text: any

text: k

text: the

text: map

text: Θ

text: restricted

text: to

text: O+(x)

text: S

text: maps

text: O+(x)

text: S

text: into

text: k

text: O+(x)

text: S.

text: Proof.

text: Let

text: y

text: O+(x)

text: S,

text: so

text: there

text: exists

text: T

text: such

text: that

text: y

text: ψ

text: (x)

text: and

text: n

text: (T

text: k.

text: As

text: k

text: T

text: x

text: in

text: Lemma

text: y

text: is

text: a

text: secondary

text: entrance

text: point.

text: Let

text: U

text: T

text: be

text: the

text: minimal

text: time

text: such

text: that

text: ψ

text: (x)

text: D+

text: D−

text: is

text: a

text: secondary

text: transition

text: point.

text: Then

text: for

text: any

text: T

text: t

text: U,

text: n

text: (t)

text: k

text: U

text: x

text: by

text: deﬁnition

text: of

text: the

text: level

text: function.

text: If

text: Θ(y)

text: is

text: deﬁned,

text: Θ(y)

text: φ

text: where

text: T

text: is

text: the

text: minimal

text: time

text: such

text: that

text: φ

text: S.

text: Such

text: a

text: point

text: is

text: a

text: secondary

text: transition

text: point,

text: so

text: in

text: particular

text: T

text: U

text: and

text: n

text: (T

text: k

text: i.e.

text: Θ(y)

text: O+(x)

text: S.

text: x

text: The

text: intersection

text: O+(η)

text: S

text: of

text: a

text: forward

text: orbit

text: surface

text: has

text: a

text: a

text: similar

text: level

text: decomposition,

text: and

text: acts

text: this

text: level

text: decomposition

text: in

text: the

text: same

text: way.