text: 2 ANTTI KA¨ENMA¨KI, TUOMAS SAHLSTEN, AND PABLO SHMERKIN One of the oldest and most fundamental concepts in analysis is that of tangent. Tangents capture the local structure of functions but are substantially better behaved. In particular, they have scaling and isotropy properties the original object lacks. One can infer global properties of the object from the collection of tangents at all points. The analogous concept for general measures is that of tangent measure introduced by Preiss [56]. These measures are the accumulation points → ∞ of the scenery µ as t . Although tangent measures share many of the good properties of x,t tangents, tangent measures do not give much information about the original object: two very diﬀerent measures may have the same set of tangent measures at every point. It may even happen for a measure to have all measures as tangent measures, at almost every point; see O’Neil [54] and Sahlsten [61]. In order to obtain a notion of tangent to a measure that captures more information, a simple but deep idea is to look at the statistics of the scenery (µ ) ≥ , rather than the collection x,t t 0 Tan(µ, x) of all the accumulation points. This leads to the notion of tangent distribution, studied (with some variations and under diﬀerent names(cid:90)) by several authors, see e.g. [71, 52, 24, 23, 35]. (cid:104) (cid:105) ≥ Tangent distributions are deﬁned to be weak accumulation points of µ , T 0, where x,T (cid:104) (cid:105) 1 T µ = δ dt. x,T T µx,t 0 That is, tangent distributions are measures on measures (or random measures), and from the dynamical point of view they are empirical measures for the scenery ﬂow. The family of all tangent T D distributions of µ at x is denoted by (µ, x). Tangent distributions describe the asymptotics of the scenery ﬂow and their support is a (possibly much smaller) subset of the set Tan(µ, x) of all tangent measures. We will see that tangent distributions capture many properties of the original object that are not invisible to averaging, such as dimension (though it should be noted that tangent distributions are blind to properties that are sensitive to changes on a sparse set of scales, such as classical rectiﬁability). The approach of this paper is to study properties of measures through the corresponding properties of their tangent distributions. Our goal is to apply ergodic theoretical methods and hence we restrict ourselves to scale invariant tangent distributions. If we additionally require that the distribution satisﬁes a suitable isotropy condition called quasi-Palm (see Deﬁnition 3.7), then such a distribution is called a fractal distribution. Roughly speaking, the quasi-Palm property guarantees that the null sets of the distribution are invariant under translation to a typical point for the measure. This kind of property turns out to be extremely useful when transferring information about a generic measure for the distribution from a single point to almost every point. A remarkable result of Hochman [24, Theorem 1.7] states that tangent distributions are indeed fractal distributions at almost every point (while the scale invariance is perhaps expected, the fact that tangent distributions satisfy the quasi-Palm property is one of the main discoveries of [24]). The geometric properties of fractals and their relation to diﬀerent notions of dimension and rectiﬁability have been an object of intensive study for several decades, both for their intrinsic interest and because of connections to other areas. In particular, questions dealing with conical densities and porosity have been studied thoroughly. The theory that started with conical density properties of Hausdorﬀ measures was later extended to more general Borel measures, which is the setting we consider. The study of porosity is motivated in part by the fact that it is a quantitative notion of singularity. Thus it is a natural problem to understand the relationship between dimensions (which are a diﬀerent way to quantify singularity) and various notions of porosity. Particular interest has been given to the study of the maximal possible dimension of a porous set or measure. This problem has received considerable attention and it has been addressed by using various methods and techniques. In this article, we show that, equipped with the machinery of fractal distributions, most of the above alluded results on conical densities and porosities are consequences of a suitable notion of rectiﬁability, and of set-theoretical porosity. In this way, we unify and explain in a coherent way a large number of previously piecemeal results. We obtain an essentially sharp and very general conical density result, and prove new results for diﬀerent kinds of porosity valid for arbitrary values ##LTLine##

text: ANTTI

text: KA¨ENMA¨KI,

text: TUOMAS

text: SAHLSTEN,

text: AND

text: PABLO

text: SHMERKIN

text: One

text: of

text: the

text: oldest

text: and

text: most

text: fundamental

text: concepts

text: in

text: analysis

text: is

text: that

text: of

text: tangent.

text: Tangents

text: capture

text: the

text: local

text: structure

text: of

text: functions

text: but

text: are

text: substantially

text: better

text: behaved.

text: In

text: particular,

text: they

text: have

text: scaling

text: and

text: isotropy

text: properties

text: the

text: original

text: object

text: lacks.

text: One

text: can

text: infer

text: global

text: properties

text: of

text: the

text: object

text: from

text: the

text: collection

text: of

text: tangents

text: at

text: all

text: points.

text: The

text: analogous

text: concept

text: for

text: general

text: measures

text: is

text: that

text: of

text: tangent

text: measure

text: introduced

text: by

text: Preiss

text: These

text: measures

text: are

text: the

text: accumulation

text: points

text: of

text: the

text: scenery

text: µ

text: as

text: t

text: Although

text: tangent

text: measures

text: share

text: many

text: of

text: the

text: good

text: properties

text: of

text: x,t

text: tangents,

text: tangent

text: measures

text: do

text: not

text: give

text: much

text: information

text: about

text: the

text: original

text: object:

text: two

text: very

text: diﬀerent

text: measures

text: may

text: have

text: the

text: same

text: set

text: of

text: tangent

text: measures

text: at

text: every

text: point.

text: It

text: may

text: even

text: happen

text: for

text: a

text: measure

text: to

text: have

text: all

text: measures

text: as

text: tangent

text: measures,

text: at

text: almost

text: every

text: point;

text: see

text: O’Neil

text: and

text: Sahlsten

text: In

text: order

text: to

text: obtain

text: a

text: notion

text: of

text: tangent

text: to

text: a

text: measure

text: that

text: captures

text: more

text: information,

text: a

text: simple

text: but

text: deep

text: idea

text: is

text: to

text: look

text: at

text: the

text: statistics

text: of

text: the

text: scenery

text: (µ

text: rather

text: than

text: the

text: collection

text: x,t

text: t

text: Tan(µ,

text: x)

text: of

text: all

text: the

text: accumulation

text: points.

text: This

text: leads

text: to

text: the

text: notion

text: of

text: tangent

text: distribution,

text: studied

text: (with

text: some

text: variations

text: and

text: under

text: diﬀerent

text: by

text: several

text: authors,

text: see

text: e.g.

text: Tangent

text: distributions

text: are

text: deﬁned

text: to

text: be

text: weak

text: accumulation

text: points

text: of

text: µ

text: T

text: where

text: x,T

text: T

text: µ

text: δ

text: dt.

text: x,T

text: T

text: µx,t

text: That

text: is,

text: tangent

text: distributions

text: are

text: measures

text: on

text: measures

text: (or

text: random

text: measures),

text: and

text: from

text: the

text: dynamical

text: point

text: of

text: view

text: they

text: are

text: empirical

text: measures

text: for

text: the

text: scenery

text: ﬂow.

text: The

text: family

text: of

text: all

text: tangent

text: T

text: D

text: distributions

text: of

text: µ

text: at

text: x

text: is

text: denoted

text: by

text: (µ,

text: x).

text: Tangent

text: distributions

text: describe

text: the

text: asymptotics

text: of

text: the

text: scenery

text: ﬂow

text: and

text: their

text: support

text: is

text: a

text: (possibly

text: much

text: smaller)

text: subset

text: of

text: the

text: set

text: Tan(µ,

text: x)

text: of

text: all

text: tangent

text: measures.

text: We

text: will

text: see

text: that

text: tangent

text: distributions

text: capture

text: many

text: properties

text: of

text: the

text: original

text: object

text: that

text: are

text: not

text: invisible

text: to

text: averaging,

text: such

text: as

text: dimension

text: (though

text: it

text: should

text: be

text: noted

text: that

text: tangent

text: distributions

text: are

text: blind

text: to

text: properties

text: that

text: are

text: sensitive

text: to

text: changes

text: on

text: a

text: sparse

text: set

text: of

text: scales,

text: such

text: as

text: classical

text: rectiﬁability).

text: The

text: approach

text: of

text: this

text: paper

text: is

text: to

text: study

text: properties

text: of

text: measures

text: through

text: the

text: corresponding

text: properties

text: of

text: their

text: tangent

text: distributions.

text: Our

text: goal

text: is

text: to

text: apply

text: ergodic

text: theoretical

text: methods

text: and

text: hence

text: we

text: restrict

text: ourselves

text: to

text: scale

text: invariant

text: tangent

text: distributions.

text: If

text: we

text: additionally

text: require

text: that

text: the

text: distribution

text: satisﬁes

text: a

text: suitable

text: isotropy

text: condition

text: called

text: quasi-Palm

text: (see

text: Deﬁnition

text: then

text: such

text: a

text: distribution

text: is

text: called

text: a

text: fractal

text: distribution.

text: Roughly

text: speaking,

text: the

text: quasi-Palm

text: property

text: guarantees

text: that

text: the

text: sets

text: of

text: the

text: distribution

text: are

text: invariant

text: under

text: translation

text: to

text: a

text: typical

text: point

text: for

text: the

text: measure.

text: This

text: kind

text: of

text: property

text: turns

text: out

text: to

text: be

text: extremely

text: useful

text: when

text: transferring

text: information

text: about

text: a

text: generic

text: measure

text: for

text: the

text: distribution

text: from

text: a

text: single

text: point

text: to

text: almost

text: every

text: point.

text: A

text: remarkable

text: result

text: of

text: Hochman

text: Theorem

text: states

text: that

text: tangent

text: distributions

text: are

text: indeed

text: fractal

text: distributions

text: at

text: almost

text: every

text: point

text: (while

text: the

text: scale

text: invariance

text: is

text: perhaps

text: expected,

text: the

text: fact

text: that

text: tangent

text: distributions

text: satisfy

text: the

text: quasi-Palm

text: property

text: is

text: one

text: of

text: the

text: main

text: discoveries

text: of

text: The

text: geometric

text: properties

text: of

text: fractals

text: and

text: their

text: relation

text: to

text: diﬀerent

text: notions

text: of

text: dimension

text: and

text: rectiﬁability

text: have

text: been

text: an

text: object

text: of

text: intensive

text: study

text: for

text: several

text: decades,

text: both

text: for

text: their

text: intrinsic

text: interest

text: and

text: because

text: of

text: connections

text: to

text: other

text: areas.

text: In

text: particular,

text: questions

text: dealing

text: with

text: conical

text: densities

text: and

text: porosity

text: have

text: been

text: studied

text: thoroughly.

text: The

text: theory

text: that

text: started

text: with

text: conical

text: density

text: properties

text: of

text: Hausdorﬀ

text: measures

text: was

text: later

text: extended

text: to

text: more

text: general

text: Borel

text: measures,

text: which

text: is

text: the

text: setting

text: we

text: consider.

text: The

text: study

text: of

text: porosity

text: is

text: motivated

text: in

text: part

text: by

text: the

text: fact

text: that

text: it

text: is

text: a

text: quantitative

text: notion

text: of

text: singularity.

text: Thus

text: it

text: is

text: a

text: natural

text: problem

text: to

text: understand

text: the

text: relationship

text: between

text: dimensions

text: (which

text: are

text: a

text: diﬀerent

text: way

text: to

text: quantify

text: singularity)

text: and

text: various

text: notions

text: of

text: porosity.

text: Particular

text: interest

text: has

text: been

text: given

text: to

text: the

text: study

text: of

text: the

text: maximal

text: possible

text: dimension

text: of

text: a

text: porous

text: set

text: or

text: measure.

text: This

text: problem

text: has

text: received

text: considerable

text: attention

text: and

text: it

text: has

text: been

text: addressed

text: by

text: using

text: various

text: methods

text: and

text: techniques.

text: In

text: this

text: article,

text: we

text: show

text: that,

text: equipped

text: with

text: the

text: machinery

text: of

text: fractal

text: distributions,

text: most

text: of

text: the

text: above

text: alluded

text: results

text: on

text: conical

text: densities

text: and

text: porosities

text: are

text: consequences

text: of

text: a

text: suitable

text: notion

text: of

text: rectiﬁability,

text: and

text: of

text: set-theoretical

text: porosity.

text: In

text: this

text: way,

text: we

text: unify

text: and

text: explain

text: in

text: a

text: coherent

text: way

text: a

text: large

text: number

text: of

text: previously

text: piecemeal

text: results.

text: We

text: obtain

text: an

text: essentially

text: sharp

text: and

text: very

text: general

text: conical

text: density

text: result,

text: and

text: prove

text: new

text: results

text: for

text: diﬀerent

text: kinds

text: of

text: porosity

text: valid

text: for

text: arbitrary

text: values

text: ##LTLine##


text: SCENERY FLOW AND GEOMETRY OF MEASURES 13 Figure 2. The center y of the α-hole of E or µ can only be chosen from the annulus A(x, cr, r). ≤ − Deﬁnition 2.22 (Annular porosity). Let 0 < α, (cid:37) 1 and write c = (1 + (cid:37)) 1. Let A(x, cr, r) be \ the closed annulus B(x, r) B(x, cr). We say that E is (cid:37)-annular α-porous at the point x and scale ∈ r, if there exists y A(x, cr, r) with | − | ∩ ∅ B(y, α(cid:37) x y ) E = . Moreover, we say that E is (cid:37)-annular α-porous at x if this happens for all small enough r > 0 and ∈ E is (cid:37)-annular α-porous if it is (cid:37)-annular α-porous at every x E. If µ is a Radon measure and ε > 0, then µ is (cid:37)-annular α-porous at the point x and scale r with ∈ threshold ε, if there exists y A(x, cr, r) with | − | ≤ µ(B(y, α(cid:37) x y )) εµ(B(x, r)). If for all small enough ε > 0 this happens for all small enough r > 0 (depending on ε), then µ is (cid:37)-annular α-porous at x and if µ is (cid:37)-annular α-porous at µ almost every x, then µ is (cid:37)-annular α-porous. Finally, mean annular porosity for sets and measures is deﬁned analogously to the classical case. Koskela and Rohde found bounds for the packing dimension of mean annular porous sets. In [38, ⊂ R Theorem 2.1], they proved that if E d is a p-mean (cid:37)-annular α-porous set, then ≤ − − dim E d Cp(cid:37)d 1αd (2.12) p where C depends only on d. They also showed that the exponents in the estimate are the best possible ones. If (cid:37) is close to zero, then also the width of the annulus A(x, cr, r) is close to zero. Thus for small α the annular porosity requires that each ball contains a lot of pores roughly of the same size. Therefore, if α is ﬁxed, we get better asymptotic behavior for the dimension of (cid:37)-annular α-porous sets. As a particular application, Koskela and Rohde showed in [38, Corollary → R 3.2] that if f : B(0, 1) d is K-quasiconformal and (cid:37)-H¨older continuous, then the Minkowski − − − dimension of f(Sd 1) is at most d C(cid:37)d 1 where C depends only on d and K. For basic properties of quasiconformal mappings, the reader is referred to the book of Ahlfors [1]. For measures, such bounds have not yet been found. We again reduce the problem back to the set case. We write ∆(cid:37)(α) := ∆(cid:37)(α) = sup{dim µ : µ is (cid:37)-annular α-porous on Rd} d p and ∆(cid:37)(α, p) := ∆(cid:37)(α, p) = sup{dim µ : µ is p-mean (cid:37)-annular α-porous on Rd}. d p The following result relates the dimension of p-mean (cid:37)-annular α-porous measures back to the dimensions of p-mean (cid:37)-annular α-porous measures of sets, thereby extending the estimate (2.12) of Koskela and Rohde from sets to measures. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SCENERY

label: paragraph

text: FLOW

text: AND

text: GEOMETRY

text: OF

text: MEASURES

text: Figure

text: The

label: caption

text: center

text: y

text: of

text: the

text: α-hole

text: of

text: E

text: or

text: µ

text: can

text: only

text: be

text: chosen

text: from

text: the

text: annulus

text: A(x,

text: cr,

text: r).

text: Deﬁnition

text: (Annular

text: porosity).

text: Let

text: α,

text: and

text: write

text: c

text: Let

text: A(x,

text: cr,

text: r)

text: be

text: the

text: closed

text: annulus

text: B(x,

text: r)

text: B(x,

text: cr).

text: We

text: say

text: that

text: E

text: is

text: α-porous

text: at

text: the

text: point

text: x

text: and

text: scale

text: r,

text: if

text: there

text: exists

text: y

text: A(x,

text: cr,

text: r)

text: with

text: B(y,

text: x

text: y

text: E

text: Moreover,

text: we

text: say

text: that

text: E

text: is

text: α-porous

text: at

text: x

text: if

text: this

text: happens

text: for

text: all

text: small

text: enough

text: r

text: and

text: E

text: is

text: α-porous

text: if

text: it

text: is

text: α-porous

text: at

text: every

text: x

text: E.

text: If

text: µ

text: is

text: a

text: Radon

text: measure

text: and

text: ε

text: then

text: µ

text: is

text: α-porous

text: at

text: the

text: point

text: x

text: and

text: scale

text: r

text: with

text: threshold

text: ε,

text: if

text: there

text: exists

text: y

text: A(x,

text: cr,

text: r)

text: with

text: µ(B(y,

text: x

text: y

text: εµ(B(x,

text: r)).

text: If

text: for

text: all

text: small

text: enough

text: ε

text: this

text: happens

text: for

text: all

text: small

text: enough

text: r

text: (depending

text: on

text: ε),

text: then

text: µ

text: is

text: α-porous

text: at

text: x

text: and

text: if

text: µ

text: is

text: α-porous

text: at

text: µ

text: almost

text: every

text: x,

text: then

text: µ

text: is

text: α-porous.

text: Finally,

text: mean

text: annular

text: porosity

text: for

text: sets

text: and

text: measures

text: is

text: deﬁned

text: analogously

text: to

text: the

text: classical

text: case.

text: Koskela

text: and

text: Rohde

text: found

text: bounds

text: for

text: the

text: packing

text: dimension

text: of

text: mean

text: annular

text: porous

text: sets.

text: In

text: R

text: Theorem

text: they

text: proved

text: that

text: if

text: E

text: d

text: is

text: a

text: p-mean

text: α-porous

text: set,

text: then

text: dim

text: E

text: d

text: p

text: where

text: C

text: depends

text: only

text: on

text: d.

text: They

text: also

text: showed

text: that

text: the

text: exponents

text: in

text: the

text: estimate

text: are

text: the

text: best

text: possible

text: ones.

text: If

text: is

text: close

text: to

text: zero,

text: then

text: also

text: the

text: width

text: of

text: the

text: annulus

text: A(x,

text: cr,

text: r)

text: is

text: close

text: to

text: zero.

text: Thus

text: for

text: small

text: α

text: the

text: annular

text: porosity

text: requires

text: that

text: each

text: ball

text: contains

text: a

text: lot

text: of

text: pores

text: roughly

text: of

text: the

text: same

text: size.

text: Therefore,

text: if

text: α

text: is

text: ﬁxed,

text: we

text: get

text: better

text: asymptotic

text: behavior

text: for

text: the

text: dimension

text: of

text: α-porous

text: sets.

text: As

text: a

text: particular

text: application,

text: Koskela

text: and

text: Rohde

text: showed

text: in

text: Corollary

text: R

text: that

text: if

text: f

text: d

text: is

text: K-quasiconformal

text: and

text: continuous,

text: then

text: the

text: Minkowski

text: dimension

text: of

text: f(Sd

text: is

text: at

text: most

text: d

text: where

text: C

text: depends

text: only

text: on

text: d

text: and

text: K.

text: For

text: basic

text: properties

text: of

text: quasiconformal

text: mappings,

text: the

text: reader

text: is

text: referred

text: to

text: the

text: book

text: of

text: Ahlfors

text: For

text: measures,

text: such

text: bounds

text: have

text: not

text: yet

text: been

text: found.

text: We

text: again

text: reduce

text: the

text: problem

text: back

text: to

text: the

text: set

text: case.

text: We

text: write

text: sup{dim

text: µ

text: µ

text: is

text: α-porous

text: on

text: Rd}

text: d

text: p

text: and

text: p)

text: p)

text: sup{dim

text: µ

text: µ

text: is

text: p-mean

text: α-porous

text: on

text: Rd}.

text: d

text: p

text: The

text: following

text: result

text: relates

text: the

text: dimension

text: of

text: p-mean

text: α-porous

text: measures

text: back

text: to

text: the

text: dimensions

text: of

text: p-mean

text: α-porous

text: measures

text: of

text: sets,

text: thereby

text: extending

text: the

text: estimate

text: of

text: Koskela

text: and

text: Rohde

text: from

text: sets

text: to

text: measures.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTFigure##

text: ##LTFigure##

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text: ##LTFigure##

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text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTFigure##

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text: 24 ANTTI KA¨ENMA¨KI, TUOMAS SAHLSTEN, AND PABLO SHMERKIN ∈ − ∈ − H for all V G(d, d k) and θ Sd 1, and this does not hold for . Thus by the weak convergence W − (cid:104) (cid:105) A ≥ A s k lim inf µ ( c) P ( c) = − . (4.2) T→∞ x,T ε ε d k ∈ A → ∈ − ∈ − Moreover, if ν c with ν ν, then for any V G(d, d k) and θ Sd 1 the weak i ε i convergence also gives \ ≥ \ ≥ ν(X(0, 1, V, α) H(0, θ, α)) lim sup ν (X(0, 1, V, α) H(0, θ, α)) ε. →∞ i i A Therefore, the closure of the set c is ε A { ∈ M \ ≥ ∈ − ∈ − } c = ν : ν(X(0, 1, V, α) H(0, θ, α)) ε for all V G(d, d k) and θ Sd 1 . (4.3) ε 1 This implies − (cid:104) (cid:105) A ≤ (cid:104) (cid:105) A ≤ A s k lim sup µ ( c) lim sup µ ( c) P ( c) = − , (4.4) →∞ x,T ε →∞ x,T ε ε d k T T L \ ≥ ∈ − ∈ − since d(X(0, 1, V, α) H(0, θ, α)) ε(d, k, α) > ε for all V G(d, d k) and θ Sd 1, and H this does not hold for . Now (4.2) and (4.4) together show that (2.2) holds with lim →∞ and − W − T the limit equals (s k)/(d k). To ﬁnish the proof, we are left to consider the case ε > ε(d, k, α). The claim follows almost immediately from the choices of ε(d, k, α) and the measure µ. Indeed, (cid:104) (cid:105) A ≤ A lim sup µ ( c) P ( c) = 0 →∞ x,T ε ε T ∈ − ∈ − L \ since there exist V G(d, d k) and θ Sd 1 such that d(X(0, 1, V, α) H(0, θ, α)) = ε(d, k, α). (cid:3) In this section we prove Theorem 2.11, which shows that average unrectiﬁability is also a suﬃcient condition to guarantee that the measure is scattered enough in the sense of conical densities, and Theorem 2.13, which provides a converse under an additional assumption on the k-dimensional densities of µ. Proof of Theorem 2.11. We begin the proof by showing that it suﬃces to prove the statement on a set of positive µ measure. Let A be the set of points x where the statement fails. If A does not have zero µ measure, let ν = µ . By Theorem 3.16, the hypothesis holds also for ν, so there is a ⊂ A set B A of positive ν measure (so also of positive µ measure) where the statement holds for ν. ∈ But this is a contradiction since for x B, − \ | − \ | − µ(X(x, e t, V, α) H(x, θ, α)) ≥ µ (X(x, e t, V, α) H(x, θ, α)) µ (B(x, e t)) − A | − A − . µ(B(x, e t)) µ (B(x, e t)) µ(B(x, e t)) A whence, thanks to the Besicovitch density point theorem [46, Corollary 2.14], the statement holds ∈ also for µ almost all x B. ≤ Now suppose to the contrary that a p-average k-unrectiﬁable measure µ and 0 < α 1 are ≤ (cid:84) such that for each 0 < ε 1 the condition (2.4) fails to hold in a set E of full µ measure. ε ∈ Recalling Theorem 3.10, we may assume that all tangent distributions at points x E are fractal { ∈ M } − ∈ ε distributions and satisfy P ( ν : spt ν is strongly k-rectiﬁable ) < 1 p. Let x E , where 1 ≤ ε the intersection is over all rational 0 < ε 1. Then (2.4) fails at x and (cid:104) (cid:105) A − lim sup µ ( ) > 1 p →∞ x,T ε T ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ANTTI

text: KA¨ENMA¨KI,

text: TUOMAS

text: SAHLSTEN,

text: AND

text: PABLO

text: SHMERKIN

text: H

text: for

text: all

text: V

text: G(d,

text: d

text: k)

text: and

text: θ

text: Sd

text: and

text: this

text: does

text: not

text: hold

text: for

text: Thus

text: by

text: the

text: weak

text: convergence

text: W

label: equation

text: A

text: A

text: s

text: k

text: lim

text: inf

text: µ

text: c)

text: P

text: c)

text: T→∞

text: x,T

text: ε

text: ε

text: d

text: k

text: A

text: Moreover,

text: if

text: ν

text: c

text: with

text: ν

text: ν,

text: then

text: for

text: any

text: V

text: G(d,

text: d

text: k)

text: and

text: θ

text: Sd

text: the

text: weak

text: i

text: ε

text: i

text: convergence

text: also

text: gives

text: V,

text: α)

text: θ,

text: α))

text: lim

text: sup

text: ν

text: V,

text: α)

text: θ,

text: α))

text: ε.

text: i

text: i

text: A

text: Therefore,

text: the

text: closure

text: of

text: the

text: set

text: c

text: is

text: ε

text: A

text: M

text: c

text: ν

text: V,

text: α)

text: θ,

text: α))

text: ε

text: for

text: all

text: V

text: G(d,

text: d

text: k)

text: and

text: θ

text: Sd

text: ε

text: This

text: implies

text: A

text: A

text: A

text: s

text: k

text: lim

text: sup

text: µ

text: c)

text: lim

text: sup

text: µ

text: c)

text: P

text: c)

text: x,T

text: ε

text: x,T

text: ε

text: ε

text: d

text: k

text: T

text: T

text: L

text: since

text: V,

text: α)

text: θ,

text: α))

text: ε(d,

text: k,

text: α)

text: ε

text: for

text: all

text: V

text: G(d,

text: d

text: k)

text: and

text: θ

text: Sd

text: and

text: H

text: this

text: does

text: not

text: hold

text: for

text: Now

text: and

text: together

text: show

text: that

text: holds

text: with

text: lim

text: and

text: W

text: T

text: the

text: limit

text: equals

text: (s

text: k)/(d

text: k).

text: To

text: ﬁnish

text: the

text: proof,

text: we

text: are

text: left

text: to

text: consider

text: the

text: case

text: ε

text: ε(d,

text: k,

text: α).

text: The

text: claim

text: follows

text: almost

text: immediately

text: from

text: the

text: choices

text: of

text: ε(d,

text: k,

text: α)

text: and

text: the

text: measure

text: µ.

text: Indeed,

text: A

text: A

text: lim

text: sup

text: µ

text: c)

text: P

text: c)

text: x,T

text: ε

text: ε

text: T

text: L

text: since

text: there

text: exist

text: V

text: G(d,

text: d

text: k)

text: and

text: θ

text: Sd

text: such

text: that

text: V,

text: α)

text: θ,

text: α))

text: ε(d,

text: k,

text: α).

text: In

text: this

text: section

text: we

text: prove

text: Theorem

text: which

text: shows

text: that

text: average

text: unrectiﬁability

text: is

text: also

text: a

text: suﬃcient

text: condition

text: to

text: guarantee

text: that

text: the

text: measure

text: is

text: scattered

text: enough

text: in

text: the

text: sense

text: of

text: conical

text: densities,

text: and

text: Theorem

text: which

text: provides

text: a

text: converse

text: under

text: an

text: additional

text: assumption

text: on

text: the

text: k-dimensional

text: densities

text: of

text: µ.

text: Proof

text: of

text: Theorem

text: We

text: begin

text: the

text: proof

text: by

text: showing

text: that

text: it

text: suﬃces

text: to

text: prove

text: the

text: statement

text: on

text: a

text: set

text: of

text: positive

text: µ

text: measure.

text: Let

text: A

text: be

text: the

text: set

text: of

text: points

text: x

text: where

text: the

text: statement

text: fails.

text: If

text: A

text: does

text: not

text: have

text: zero

text: µ

text: measure,

text: let

text: ν

text: µ

text: By

text: Theorem

text: the

text: hypothesis

text: holds

text: also

text: for

text: ν,

text: so

text: there

text: is

text: a

text: A

text: set

text: B

text: A

text: of

text: positive

text: ν

text: measure

text: (so

text: also

text: of

text: positive

text: µ

text: measure)

text: where

text: the

text: statement

text: holds

text: for

text: ν.

text: But

text: this

text: is

text: a

text: contradiction

text: since

text: for

text: x

text: B,

text: µ(X(x,

text: e

text: t,

text: V,

text: α)

text: H(x,

text: θ,

text: α))

text: µ

text: (X(x,

text: e

text: t,

text: V,

text: α)

text: H(x,

text: θ,

text: α))

text: µ

text: (B(x,

text: e

text: t))

text: A

text: A

text: µ(B(x,

text: e

text: t))

text: µ

text: (B(x,

text: e

text: t))

text: µ(B(x,

text: e

text: t))

text: A

text: whence,

text: thanks

text: to

text: the

text: Besicovitch

text: density

text: point

text: theorem

text: Corollary

text: the

text: statement

text: holds

text: also

text: for

text: µ

text: almost

text: all

text: x

text: B.

text: Now

text: suppose

text: to

text: the

text: contrary

text: that

text: a

text: p-average

text: k-unrectiﬁable

text: measure

text: µ

text: and

text: α

text: are

text: such

text: that

text: for

text: each

text: ε

text: the

text: condition

text: fails

text: to

text: hold

text: in

text: a

text: set

text: E

text: of

text: full

text: µ

text: measure.

text: ε

text: Recalling

text: Theorem

text: we

text: may

text: assume

text: that

text: all

text: tangent

text: distributions

text: at

text: points

text: x

text: E

text: are

text: fractal

text: M

text: ε

text: distributions

text: and

text: satisfy

text: P

text: ν

text: spt

text: ν

text: is

text: strongly

text: k-rectiﬁable

text: p.

text: Let

text: x

text: E

text: where

text: ε

text: the

text: intersection

text: is

text: over

text: all

text: rational

text: ε

text: Then

text: fails

text: at

text: x

text: and

text: A

text: lim

text: sup

text: µ

text: p

text: x,T

text: ε

text: T

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 30 ANTTI KA¨ENMA¨KI, TUOMAS SAHLSTEN, AND PABLO SHMERKIN ∈ T D ≤ and there exists a fractal distribution P (µ, x) with dim (µ, x) dim P + δ/3. This is A loc possible by Theorem 3.21. Since µ is α-porous at x and is closed we have ε A ≥ (cid:104) (cid:105) A ≥ P ( ) lim inf µ ( ) 1 ε →∞ x,T ε T for all ε > 0. Let − Q = pP + (1 p)δLd. According to Theorem 3.15(cid:90), there exists a uniformly scaling measure ν that generates Q. It follows from Deﬁnition 3.18 and Theorem 3.21 that ν is exact dimensional with − ≥ − − dim ν = dim Q = dim µ dQ(ν) = p dim P + (1 p)d p(∆(α) δ) + (1 p)d. Observe also that A A − A ≥ Q( ε) = pP ( ε) + (1 p)δLd( ε) p for all ε > 0. Recalling Lemma 4.9 and Theorem 3.22, we conclude that (cid:104) (cid:105) A ≥ (cid:104) (cid:105) U ≥ U ≥ A ≥ lim inf ν ( ) lim inf ν ( ) Q( ) Q( ) p →∞ z,T 2ε →∞ z,T 2ε 2ε ε T T for all ε > 0 at ν almost every z, that is, ν is p-mean α-porous. The proof is ﬁnished by letting ↓ (cid:3) δ 0. In this section, we prove Theorem 2.23 which shows the results for annular porosity corresponding to Theorems 2.16 and 2.20. Investigating the proofs of Theorems 2.16 and 2.20, we see that the geometric information obtained from the deﬁnition of porosity is only used to show the required A U properties of the sets and . The corresponding sets in the annular porosity case are ε ε A◦ { ∈ M | | ≤ ∈ } := ν : ν(B(y, α(cid:37) y )) ε for some y A(0, c, 1) ε 1 and U◦ { ∈ M | | ∈ } := ν : ν(B(y, α(cid:37) y )) < ε for some y A(0, c, 1) . ε 1 A◦ U◦ To prove Theorem 2.23, it in fact suﬃces to show that is closed and is open. Since the proof ε ε for the openness is the same as that of Lemma 4.9 we just verify the closedness. A◦ ≥ Lemma 4.10. The set is closed for all ε 0. ε A◦ ∈ M → Proof. Suppose that there are a sequence (ν ) of measures in and ν so that ν ν weakly. (cid:48) ∈ i i | | ≤ ε 1 i Fix α < α. Choose y A(0, c, 1) with ν (B(y , α(cid:37) y )) ε. By the compactness of A(0, c, 1), after i i i i → ∈ (cid:48) possibly passing to a subsequence, we may assume that y y A(0, c, 1). Since α < α, we have (cid:48) | | ⊂ | | i | | B(y, α (cid:37) y ) B(y , α(cid:37) y ) for all large enough i. Thus by the openness of B(y, α(cid:37) y ), we have i i (cid:48) | | ≤ (cid:48) | | ≤ | | ≤ ν(B(y, α (cid:37) y )) lim inf ν (B(y, α (cid:37) y )) lim inf ν (B(y , α(cid:37) y )) ε. →∞ i →∞ i i i i i (cid:48) ↑ | | ≤ ∈ A◦ A◦ (cid:3) Letting α α gives ν(B(y, α(cid:37) y )) ε. In particular, ν , showing that is closed. ε ε ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ANTTI

text: KA¨ENMA¨KI,

text: TUOMAS

text: SAHLSTEN,

text: AND

text: PABLO

text: SHMERKIN

text: T

text: D

text: and

text: there

text: exists

text: a

text: fractal

text: distribution

text: P

text: (µ,

text: x)

text: with

text: dim

text: (µ,

text: x)

text: dim

text: P

text: This

text: is

text: A

text: loc

text: possible

text: by

text: Theorem

text: Since

text: µ

text: is

text: α-porous

text: at

text: x

text: and

text: is

text: closed

text: we

text: have

text: ε

text: A

text: A

text: P

text: lim

text: inf

text: µ

text: ε

text: x,T

text: ε

text: T

text: for

text: all

text: ε

text: Let

text: Q

text: pP

text: p)δLd.

text: According

text: to

text: Theorem

text: there

text: exists

text: a

text: uniformly

text: scaling

text: measure

text: ν

text: that

text: generates

text: Q.

text: It

text: follows

text: from

text: Deﬁnition

text: and

text: Theorem

text: that

text: ν

text: is

text: exact

text: dimensional

text: with

text: dim

text: ν

text: dim

text: Q

text: dim

text: µ

text: dQ(ν)

text: p

text: dim

text: P

text: p)d

text: p(∆(α)

text: δ)

text: p)d.

text: Observe

text: also

text: that

text: A

text: A

text: A

text: Q(

text: ε)

text: pP

text: ε)

text: p)δLd(

text: ε)

text: p

text: for

text: all

text: ε

text: Recalling

text: Lemma

text: and

text: Theorem

text: we

text: conclude

text: that

text: A

text: U

text: U

text: A

text: lim

text: inf

text: ν

text: lim

text: inf

text: ν

text: Q(

text: Q(

text: p

text: z,T

text: 2ε

text: z,T

text: 2ε

text: 2ε

text: ε

text: T

text: T

text: for

text: all

text: ε

text: at

text: ν

text: almost

text: every

text: z,

text: that

text: is,

text: ν

text: is

text: p-mean

text: α-porous.

text: The

text: proof

text: is

text: ﬁnished

text: by

text: letting

text: δ

text: In

text: this

text: section,

text: we

text: prove

text: Theorem

text: which

text: shows

text: the

text: results

text: for

text: annular

text: porosity

text: corresponding

text: to

text: Theorems

text: and

text: Investigating

text: the

text: proofs

text: of

text: Theorems

text: and

text: we

text: see

text: that

text: the

text: geometric

text: information

text: obtained

text: from

text: the

text: deﬁnition

text: of

text: porosity

text: is

text: only

text: used

text: to

text: show

text: the

text: required

text: A

text: U

text: properties

text: of

text: the

text: sets

text: and

text: The

text: corresponding

text: sets

text: in

text: the

text: annular

text: porosity

text: case

text: are

text: ε

text: ε

text: A◦

text: M

text: ν

text: ν(B(y,

text: y

text: ε

text: for

text: some

text: y

text: c,

text: ε

text: and

text: U◦

text: M

text: ν

text: ν(B(y,

text: y

text: ε

text: for

text: some

text: y

text: c,

text: ε

text: A◦

text: U◦

text: To

text: prove

text: Theorem

text: it

text: in

text: fact

text: suﬃces

text: to

text: show

text: that

text: is

text: closed

text: and

text: is

text: open.

text: Since

text: the

text: proof

text: ε

text: ε

text: for

text: the

text: openness

text: is

text: the

text: same

text: as

text: that

text: of

text: Lemma

text: we

text: just

text: verify

text: the

text: closedness.

text: A◦

text: Lemma

text: The

text: set

text: is

text: closed

text: for

text: all

text: ε

text: ε

text: A◦

text: M

text: Proof.

text: Suppose

text: that

text: there

text: are

text: a

text: sequence

text: (ν

text: of

text: measures

text: in

text: and

text: ν

text: so

text: that

text: ν

text: ν

text: weakly.

text: i

text: i

text: ε

text: i

text: Fix

text: α

text: α.

text: Choose

text: y

text: c,

text: with

text: ν

text: (B(y

text: y

text: ε.

text: By

text: the

text: compactness

text: of

text: c,

text: after

text: i

text: i

text: i

text: i

text: possibly

text: passing

text: to

text: a

text: subsequence,

text: we

text: may

text: assume

text: that

text: y

text: y

text: c,

text: Since

text: α

text: α,

text: we

text: have

text: i

text: B(y,

text: α

text: y

text: B(y

text: y

text: for

text: all

text: large

text: enough

text: i.

text: Thus

text: by

text: the

text: openness

text: of

text: B(y,

text: y

text: we

text: have

text: i

text: i

text: ν(B(y,

text: α

text: y

text: lim

text: inf

text: ν

text: (B(y,

text: α

text: y

text: lim

text: inf

text: ν

text: (B(y

text: y

text: ε.

text: i

text: i

text: i

text: i

text: i

text: i

text: A◦

text: A◦

text: Letting

text: α

text: α

text: gives

text: ν(B(y,

text: y

text: ε.

text: In

text: particular,

text: ν

text: showing

text: that

text: is

text: closed.

text: ε

text: ε

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 ANTTI KA¨ENMA¨KI, TUOMAS SAHLSTEN, AND PABLO SHMERKIN regardless of whether the inequalities in their deﬁnitions are strict or not. Moreover, since for each − x there are at most countably many t such that the boundary of B(x, e t) has positive µ-mass, the result also holds if in the deﬁnition of X(x, r, V, α) we intersect with the closed ball B(x, r) − instead. Likewise, in (2.2) we divide by the mass of the closed ball B(x, e t) in order to match − ≤ − the deﬁnition of the scenery ﬂow but, since µ(B(x, e t)) µ(B(x, e t)), the result also holds if we − divide by µ(B(x, e t)) instead. These observations also apply to Theorem 2.11 below. The proof of Theorem 2.3 is based on showing that there cannot be “too many” rectiﬁable tangent measures. Let us next give the precise deﬁnition for rectiﬁability and start examining its relationship to conical density results in more detail. Deﬁnition 2.5 (Rectiﬁability of sets an(cid:16)d me(cid:91)asures). (cid:17)A set E ⊂ Rd is called k-rectiﬁable if there R → R are countably many Lipschitz maps f : k d so that i H \ R k E f ( k) = 0. i i (cid:28) H Moreover, we say that a Radon measure ν is k-rectiﬁable if ν k and there exists a k-rectiﬁable ⊂ R R \ set E d such that ν( d E) = 0 A Radon measure µ is purely k-unrectiﬁable if it gives no mass to k-rectiﬁable sets and E is H | purely k-unrectiﬁable if the restriction k is purely k-unrectiﬁable. E While the deﬁnition above is nowadays standard and very useful in dealing with properties valid H up to sets of zero k measure (see e.g. [46, Chapters 15–19]), we will also require a variant which corresponds to the deﬁnition of rectiﬁability in Federer [19, 3.2.14]. To avoid any confusion, we call it strong rectiﬁability. ⊂ R Deﬁnition 2.6 (Strong rectiﬁability of sets). (cid:91)A set E d is called strongly k-rectiﬁable if there R → R exist countably many Lipschitz maps f : k d such that i ⊂ R E f ( k). i i Remark 2.7. A strongly k-rectiﬁable set is obviously k-rectiﬁable. On the other hand, there are ⊂ R many sets which are k-rectiﬁable but not strongly k-rectiﬁable, for example, any set E d such H that k(E) = 0 but dim E > k. p Pure unrectiﬁability is also a condition which should guarantee that the measure is scattered in many directions. Indeed, under suitable assumption, it leads to a conical density result: for example, the following is proved in K¨aenm¨aki [33, Theorem 8]. ≤ Theorem 2.8. If M > 0 and 0 < α 1, then there is a constant ε > 0 depending only on M ∈ N ∈ { − } ∈ − and α satisfying the following: For every d , k 1, . . . , d 1 , V G(d, d k), and purely R k-unrectiﬁable measure µ on d with µ(B(x, 2r)) lim sup < M (2.3) ↓ µ(B(x, r)) r 0 ∈ R at µ almost every x d, it holds that µ(X(x, r, V, α)) ≥ lim sup ε ↓ µ(B(x, r)) r 0 ∈ R at µ almost every x d. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ANTTI

text: KA¨ENMA¨KI,

text: TUOMAS

text: SAHLSTEN,

text: AND

text: PABLO

text: SHMERKIN

text: regardless

text: of

text: whether

text: the

text: inequalities

text: in

text: their

text: deﬁnitions

text: are

text: strict

text: or

text: not.

text: Moreover,

text: since

text: for

text: each

text: x

text: there

text: are

text: at

text: most

text: countably

text: many

text: t

text: such

text: that

text: the

text: boundary

text: of

text: B(x,

text: e

text: t)

text: has

text: positive

text: µ-mass,

text: the

text: result

text: also

text: holds

text: if

text: in

text: the

text: deﬁnition

text: of

text: X(x,

text: r,

text: V,

text: α)

text: we

text: intersect

text: with

text: the

text: closed

text: ball

text: B(x,

text: r)

text: instead.

text: Likewise,

text: in

text: we

text: divide

text: by

text: the

text: mass

text: of

text: the

text: closed

text: ball

text: B(x,

text: e

text: t)

text: in

text: order

text: to

text: match

text: the

text: deﬁnition

text: of

text: the

text: scenery

text: ﬂow

text: but,

text: since

text: µ(B(x,

text: e

text: t))

text: µ(B(x,

text: e

text: t)),

text: the

text: result

text: also

text: holds

text: if

text: we

text: divide

text: by

text: µ(B(x,

text: e

text: t))

text: instead.

text: These

text: observations

text: also

text: apply

text: to

text: Theorem

text: below.

text: The

text: proof

text: of

text: Theorem

text: is

text: based

text: on

text: showing

text: that

text: there

text: cannot

text: be

text: “too

text: many”

text: rectiﬁable

text: tangent

text: measures.

text: Let

text: us

text: next

text: give

text: the

text: precise

text: deﬁnition

text: for

text: rectiﬁability

text: and

text: start

text: examining

text: its

text: relationship

text: to

text: conical

text: density

text: results

text: in

text: more

text: detail.

text: Deﬁnition

text: (Rectiﬁability

text: of

text: sets

text: set

text: E

text: Rd

text: is

text: called

text: k-rectiﬁable

text: if

text: there

text: R

text: R

text: are

text: countably

text: many

text: Lipschitz

text: maps

text: f

text: k

text: d

text: so

text: that

text: i

text: H

text: R

text: k

text: E

text: f

text: k)

text: i

text: i

text: H

text: Moreover,

text: we

text: say

text: that

text: a

text: Radon

text: measure

text: ν

text: is

text: k-rectiﬁable

text: if

text: ν

text: k

text: and

text: there

text: exists

text: a

text: k-rectiﬁable

text: R

text: R

text: set

text: E

text: d

text: such

text: that

text: ν(

text: d

text: E)

text: A

text: Radon

text: measure

text: µ

text: is

text: purely

text: k-unrectiﬁable

text: if

text: it

text: gives

text: no

text: mass

text: to

text: k-rectiﬁable

text: sets

text: and

text: E

text: is

text: H

text: purely

text: k-unrectiﬁable

text: if

text: the

text: restriction

text: k

text: is

text: purely

text: k-unrectiﬁable.

text: E

text: While

text: the

text: deﬁnition

text: above

text: is

text: nowadays

text: standard

text: and

text: very

text: useful

text: in

text: dealing

text: with

text: properties

text: valid

text: H

text: up

text: to

text: sets

text: of

text: zero

text: k

text: measure

text: (see

text: e.g.

text: Chapters

text: we

text: will

text: also

text: require

text: a

text: variant

text: which

text: corresponds

text: to

text: the

text: deﬁnition

text: of

text: rectiﬁability

text: in

text: Federer

text: To

text: avoid

text: any

text: confusion,

text: we

text: call

text: it

text: strong

text: rectiﬁability.

text: R

text: Deﬁnition

text: (Strong

text: rectiﬁability

text: of

text: sets).

text: set

text: E

text: d

text: is

text: called

text: strongly

text: k-rectiﬁable

text: if

text: there

text: R

text: R

text: exist

text: countably

text: many

text: Lipschitz

text: maps

text: f

text: k

text: d

text: such

text: that

text: i

text: R

text: E

text: f

text: k).

text: i

text: i

text: Remark

text: A

text: strongly

text: k-rectiﬁable

text: set

text: is

text: obviously

text: k-rectiﬁable.

text: On

text: the

text: other

text: hand,

text: there

text: are

text: R

text: many

text: sets

text: which

text: are

text: k-rectiﬁable

text: but

text: not

text: strongly

text: k-rectiﬁable,

text: for

text: example,

text: any

text: set

text: E

text: d

text: such

text: H

text: that

text: k(E)

text: but

text: dim

text: E

text: k.

text: p

text: Pure

text: unrectiﬁability

text: is

text: also

text: a

text: condition

text: which

text: should

text: guarantee

text: that

text: the

text: measure

text: is

text: scattered

text: in

text: many

text: directions.

text: Indeed,

text: under

text: suitable

text: assumption,

text: it

text: leads

text: to

text: a

text: conical

text: density

text: result:

text: for

text: example,

text: the

text: following

text: is

text: proved

text: in

text: K¨aenm¨aki

text: Theorem

text: Theorem

text: If

text: M

text: and

text: α

text: then

text: there

text: is

text: a

text: constant

text: ε

text: depending

text: only

text: on

text: M

text: N

text: and

text: α

text: satisfying

text: the

text: following:

text: For

text: every

text: d

text: k

text: d

text: V

text: G(d,

text: d

text: k),

text: and

text: purely

text: R

text: k-unrectiﬁable

text: measure

text: µ

text: on

text: d

text: with

text: µ(B(x,

text: lim

text: sup

text: M

text: µ(B(x,

text: r))

text: r

text: R

text: at

text: µ

text: almost

text: every

text: x

text: d,

text: it

text: holds

text: that

text: µ(X(x,

text: r,

text: V,

text: α))

text: lim

text: sup

text: ε

text: µ(B(x,

text: r))

text: r

text: R

text: at

text: µ

text: almost

text: every

text: x

text: d.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 ANTTI KA¨ENMA¨KI, TUOMAS SAHLSTEN, AND PABLO SHMERKIN packing dimensions of a porous set or measure. See K¨aenm¨aki [32] and the surveys of J¨arvenp¨a¨a [28] and Shmerkin [66] for further background and discussion. R It is easy to see that if a set in d has positive porosity, then its dimension must be strictly less than the dimension of the ambient space. The asymptotical behavior of the dimension drop was described by Trocenko [68] and Martio and Vuorinen [43]. On the other hand, by Mattila [45] and ⊂ R Salli [64], we know that the dimension of sets E d with porosity α close to the maximum value − 1 cannot be much larger than d 1: there exists a constant c > 0 depending only on d such that 2 ≤ − c dim E d 1 + − − . (2.8) p log(1 2α) The above mentioned articles generalize these estimates further for other notions of porosity and to measures by using various methods and techniques. A consequence of our results in this article is that almost all of these generalizations follow directly from the original Hausdorﬀ dimension estimates for porous sets (which not only came earlier but are conceptually simpler to obtain). R Let us write ∆(α) for the maximal upper packing dimension of an α-porous measure on d, that is, { R } ∆(α) := ∆ (α) = sup dim µ : µ is an α-porous Radon measure on d . d p It has been observed in the earlier works [6, 34, 67, 62] that { ⊂ R } sup dim E : E d is α-porous H is, up to certain constants, asymptotically equal to ∆(α) when α is either close to its minimum or maximum value. In other words, whether we consider sets or measures, Hausdorﬀ or packing dimension, the largest possible dimension turns out to be the same. Moreover, the examples illustrating the sharpness of these results are always self-similar. Using fractal distributions we are able to show that this is a general phenomenon with a conceptual explanation. Concerning the intermediate values of α, Salli [64, Remark 3.8.2(1)] proved that, for each ≤ 0 < α 1/2, { ⊂ R } log 2 sup dim E : E is α-porous = dim C = − − − , (2.9) H H α log(2 2α) log(1 2α) ⊂ R − where C is the standard 1−2α-Cantor set. This result is intrinsic to the real line and the α 2 2α proof does not generalize to higher dimensions. Besides generalizing this to porous measures, the next result also says that largest possible dimensions obtained from α-porous sets and measures are the same for all values of α, in any dimension. In particular, the packing dimension estimates for α-porous measures follow immediately from the Hausdorﬀ dimension estimates for α-porous sets when α is either close to its minimum or maximum value. Moreover, extremal measures (that is, α-porous measures of maximal dimension) exist, and can be chosen to be uniformly scaling, which is a generalized version of self-similarity inspired by ergodic theory, see Deﬁnition 3.13 below. ≤ Theorem 2.16. For any 0 < α 1/2 we have { ⊂ R } ∆ (α) = sup dim E : E d is α-porous . d H Moreover, the supremum in the deﬁnition of ∆ (α) is attained by an exact-dimensional measure, d which furthermore is uniformly scaling. Theorem 2.16 is loosely inspired by a result of Furstenberg on galleries of sets [22, Theorem R 5.1]: a gallery is a collection of compact subsets of d which is closed under passing to subsets, under magniﬁcations, and under limits in the Hausdorﬀ metric. Furstenberg proved that for any gallery there is a measure supported on a set of the gallery whose Hausdorﬀ dimension equals the supremum of the Assouad (in particular, also Hausdorﬀ or packing) dimensions of sets in the gallery. Note that no direct application of this result is possible in our context, since porous sets ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ANTTI

text: KA¨ENMA¨KI,

text: TUOMAS

text: SAHLSTEN,

text: AND

text: PABLO

text: SHMERKIN

text: packing

text: dimensions

text: of

text: a

text: porous

text: set

text: or

text: measure.

text: See

text: K¨aenm¨aki

text: and

text: the

text: surveys

text: of

text: J¨arvenp¨a¨a

text: and

text: Shmerkin

text: for

text: further

text: background

text: and

text: discussion.

text: R

text: It

text: is

text: easy

text: to

text: see

text: that

text: if

text: a

text: set

text: in

text: d

text: has

text: positive

text: porosity,

text: then

text: its

text: dimension

text: must

text: be

text: strictly

text: less

text: than

text: the

text: dimension

text: of

text: the

text: ambient

text: space.

text: The

text: asymptotical

text: behavior

text: of

text: the

text: dimension

text: drop

text: was

text: described

text: by

text: Trocenko

text: and

text: Martio

text: and

text: Vuorinen

text: On

text: the

text: other

text: hand,

text: by

text: Mattila

text: and

text: R

text: Salli

text: we

text: know

text: that

text: the

text: dimension

text: of

text: sets

text: E

text: d

text: with

text: porosity

text: α

text: close

text: to

text: the

text: maximum

text: value

text: cannot

text: be

text: much

text: larger

text: than

text: d

text: there

text: exists

text: a

text: constant

text: c

text: depending

text: only

text: on

text: d

text: such

text: that

text: c

text: dim

text: E

text: d

text: p

text: The

text: above

text: mentioned

text: articles

text: generalize

text: these

text: estimates

text: further

text: for

text: other

text: notions

text: of

text: porosity

text: and

text: to

text: measures

text: by

text: using

text: various

text: methods

text: and

text: techniques.

text: A

text: consequence

text: of

text: our

text: results

text: in

text: this

text: article

text: is

text: that

text: almost

text: all

text: of

text: these

text: generalizations

text: follow

text: directly

text: from

text: the

text: original

text: Hausdorﬀ

text: dimension

text: estimates

text: for

text: porous

text: sets

text: (which

text: not

text: only

text: came

text: earlier

text: but

text: are

text: conceptually

text: simpler

text: to

text: obtain).

text: R

text: Let

text: us

text: write

text: ∆(α)

text: for

text: the

text: maximal

text: upper

text: packing

text: dimension

text: of

text: an

text: α-porous

text: measure

text: on

text: d,

text: that

text: is,

text: R

text: ∆(α)

text: (α)

text: sup

text: dim

text: µ

text: µ

text: is

text: an

text: α-porous

text: Radon

text: measure

text: on

text: d

text: d

text: p

text: It

text: has

text: been

text: observed

text: in

text: the

text: earlier

text: works

text: that

text: R

text: sup

text: dim

text: E

text: E

text: d

text: is

text: α-porous

text: H

text: is,

text: up

text: to

text: certain

text: constants,

text: asymptotically

text: equal

text: to

text: ∆(α)

text: when

text: α

text: is

text: either

text: close

text: to

text: its

text: minimum

text: or

text: maximum

text: value.

text: In

text: other

text: words,

text: whether

text: we

text: consider

text: sets

text: or

text: measures,

text: Hausdorﬀ

text: or

text: packing

text: dimension,

text: the

text: largest

text: possible

text: dimension

text: turns

text: out

text: to

text: be

text: the

text: same.

text: Moreover,

text: the

text: examples

text: illustrating

text: the

text: sharpness

text: of

text: these

text: results

text: are

text: always

text: self-similar.

text: Using

text: fractal

text: distributions

text: we

text: are

text: able

text: to

text: show

text: that

text: this

text: is

text: a

text: general

text: phenomenon

text: with

text: a

text: conceptual

text: explanation.

text: Concerning

text: the

text: intermediate

text: values

text: of

text: α,

text: Salli

text: Remark

text: proved

text: that,

text: for

text: each

text: α

text: R

text: log

text: sup

text: dim

text: E

text: E

text: is

text: α-porous

text: dim

text: C

text: H

text: H

text: α

text: R

text: where

text: C

text: is

text: the

text: standard

text: set.

text: This

text: result

text: is

text: intrinsic

text: to

text: the

text: real

text: line

text: and

text: the

text: α

text: 2α

text: proof

text: does

text: not

text: generalize

text: to

text: higher

text: dimensions.

text: Besides

text: generalizing

text: this

text: to

text: porous

text: measures,

text: the

text: next

text: result

text: also

text: says

text: that

text: largest

text: possible

text: dimensions

text: obtained

text: from

text: α-porous

text: sets

text: and

text: measures

text: are

text: the

text: same

text: for

text: all

text: values

text: of

text: α,

text: in

text: any

text: dimension.

text: In

text: particular,

text: the

text: packing

text: dimension

text: estimates

text: for

text: α-porous

text: measures

text: follow

text: immediately

text: from

text: the

text: Hausdorﬀ

text: dimension

text: estimates

text: for

text: α-porous

text: sets

text: when

text: α

text: is

text: either

text: close

text: to

text: its

text: minimum

text: or

text: maximum

text: value.

text: Moreover,

text: extremal

text: measures

text: (that

text: is,

text: α-porous

text: measures

text: of

text: maximal

text: dimension)

text: exist,

text: and

text: can

text: be

text: chosen

text: to

text: be

text: uniformly

text: scaling,

text: which

text: is

text: a

text: generalized

text: version

text: of

text: self-similarity

text: inspired

text: by

text: ergodic

text: theory,

text: see

text: Deﬁnition

text: below.

text: Theorem

text: For

text: any

text: α

text: we

text: have

text: R

text: (α)

text: sup

text: dim

text: E

text: E

text: d

text: is

text: α-porous

text: d

text: H

text: Moreover,

text: the

text: supremum

text: in

text: the

text: deﬁnition

text: of

text: (α)

text: is

text: attained

text: by

text: an

text: exact-dimensional

text: measure,

text: d

text: which

text: furthermore

text: is

text: uniformly

text: scaling.

text: Theorem

text: is

text: loosely

text: inspired

text: by

text: a

text: result

text: of

text: Furstenberg

text: on

text: galleries

text: of

text: sets

text: Theorem

text: R

text: a

text: gallery

text: is

text: a

text: collection

text: of

text: compact

text: subsets

text: of

text: d

text: which

text: is

text: closed

text: under

text: passing

text: to

text: subsets,

text: under

text: magniﬁcations,

text: and

text: under

text: limits

text: in

text: the

text: Hausdorﬀ

text: metric.

text: Furstenberg

text: proved

text: that

text: for

text: any

text: gallery

text: there

text: is

text: a

text: measure

text: supported

text: on

text: a

text: set

text: of

text: the

text: gallery

text: whose

text: Hausdorﬀ

text: dimension

text: equals

text: the

text: supremum

text: of

text: the

text: Assouad

text: (in

text: particular,

text: also

text: Hausdorﬀ

text: or

text: packing)

text: dimensions

text: of

text: sets

text: in

text: the

text: gallery.

text: Note

text: that

text: no

text: direct

text: application

text: of

text: this

text: result

text: is

text: possible

text: in

text: our

text: context,

text: since

text: porous

text: sets

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 P. BRANDA˜O, J. PALIS, AND V. PINHEIRO Figure 5. The map g (on the right side) is equal to f outside the interval ∈ [p , c] and has q (p , c) as a ﬁxed point. γn γn O O ∀ ∈ \ Re( +(η)) = Re( +(η)) η C C (I), (6) g f g g { ∈ O ∩ (cid:54) ∅} where C (I) = η C ; Re( +(η)) I = . C ∩ g ∅ g g − ∈ As J = (because J is a wandering interval) and as f(cid:96)(p i) C , we obtain f γn γn γn f that { ∈ O ∩ (cid:54) ∅} ∪ { } C (I) = η C ; Re( +(η)) I = p + i, γ . g (cid:18) f f (cid:19) (cid:18) γn n (cid:19) (cid:91) (cid:91) Thus, it follows from Lemma 15 that ⊂ O ∩ O ω (x) = ω (x) Re( +(α)) Re( +(α)) , (7) f g f g ∈ ∈ \ α Cn−1 α Cg Cg(I) ∈ \ \ for almost al(cid:91)l x [0, 1] B(f) = [0, 1(cid:91)] B(g) (cid:91) − ∈ If γ i C (I) then n g O O ⊂ O Re( +(α)) = Re( +(α)) Re( +(α)) g f f ∈ \ ∈ \ ∈ \{ } α Cg Cg(I) α (cid:91)Cf Cg(I) α(cid:91)Cf γn and it follows from (7) that ⊂ O O ω (x) = ω (x) Re( +(α)) = Re( +(α)), f g f f ∈ \{ } ∈ α Cn−1 γn α Cn ∈ \ \ for almost all x [0, 1] B(f) = [0, 1] B(g). − ∈ Thus, assume that γ i / C (I) and let n g { · · · − } { − · · · − − } K(g) = p , , f(cid:96) 1(p ) = Re(p i), , Re(g(cid:96) 1(p i)) . γn γn γn γn Then ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P.

text: BRANDA˜O,

text: J.

text: PALIS,

text: AND

text: V.

text: PINHEIRO

text: Figure

text: The

text: map

text: g

text: (on

text: the

text: right

text: side)

text: is

text: equal

text: to

text: f

text: outside

text: the

text: interval

text: [p

text: c]

text: and

text: has

text: q

text: (p

text: c)

text: as

text: a

text: ﬁxed

text: point.

text: γn

text: γn

text: O

text: O

text: Re(

text: +(η))

text: Re(

text: +(η))

text: η

text: C

text: C

text: (I),

text: g

text: f

text: g

text: g

text: O

font: default

text: where

text: C

text: (I)

text: η

text: C

text: Re(

text: +(η))

text: I

text: C

text: g

text: g

text: g

text: As

text: J

text: (because

text: J

text: is

text: a

text: wandering

text: interval)

text: and

text: as

text: i)

text: C

text: we

text: obtain

text: f

text: γn

text: γn

text: γn

text: f

text: that

text: O

text: C

text: (I)

text: η

text: C

text: Re(

text: +(η))

text: I

text: p

text: i,

text: γ

text: g

text: f

text: f

text: γn

text: n

text: Thus,

text: it

text: follows

text: from

text: Lemma

text: that

text: O

text: O

text: ω

text: (x)

text: ω

text: (x)

text: Re(

text: +(α))

text: Re(

text: +(α))

text: f

text: g

text: f

text: g

text: α

text: α

text: Cg

text: Cg(I)

text: for

text: almost

text: x

text: B(f)

text: B(g)

text: If

text: γ

text: i

text: C

text: (I)

text: then

text: n

text: g

text: O

text: O

text: O

text: Re(

text: +(α))

text: Re(

text: +(α))

text: Re(

text: +(α))

text: g

text: f

text: f

text: α

text: Cg

text: Cg(I)

text: α

text: Cg(I)

text: γn

text: and

text: it

text: follows

text: from

text: that

text: O

text: O

text: ω

text: (x)

text: ω

text: (x)

text: Re(

text: +(α))

text: Re(

text: +(α)),

text: f

text: g

text: f

text: f

text: α

text: γn

text: α

text: Cn

text: for

text: almost

text: all

text: x

text: B(f)

text: B(g).

text: Thus,

text: assume

text: that

text: γ

text: i

text: C

text: (I)

text: and

text: let

text: n

text: g

text: K(g)

text: p

text: Re(p

text: i),

text: i))

text: γn

text: γn

text: γn

text: γn

text: Then

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:83) 20 P. BRANDA˜O, J. PALIS, AND V. PINHEIRO ⊂ O Given K C , let A(K) = ∈ Re( +(α(cid:63))) . Thus, Lemma 17 can be rewritten as: f α K f ⊂ ∈ \ ∪ ω (x) A(C0), for almost all x [0, 1] (B (f) B (0)). (10) f f 0 1 ∈ { ∈ ∈ } ∩ Given x [0, 1], let [C ] = α C ; α(cid:63) ω (x) . Notice that [C ] = C0 [C ] . f x f f f x f f x ∈ ⊂ \ { } Lemma 18 (Parallax II). For almost all x B(f), we have ω (x) A(C γ ) for all ∈ \ f f γ C [C ] . f f x ∈ U { ∈ \ ∈ } U { ∈ Proof. Let γ C and set (γ) = x [0, 1] B(f) ; γ / ω (x) , that is, (γ) = x \ ∈ f \ } f U [0, 1] B(f) ; γ C [C ] . To prove the claim above, we may assume Leb( (γ)) > 0. f ∈ f x − ∈ C By Lemma 17, that γ C0. We can also assume that γ = c i for some c , the case ∈ f ∞ O(cid:83)− C (cid:54)(cid:51) f c + i is analogous. As γ / C , either ( ) p < c or p = γ. U { ∈ U f O ∩f f− γ ∅}γ ≥ Let (γ, n) = x (γ) ; +(x) (p 1/n, c) = and let n 1 be such that C ∩ − ∅ U f γ U 0 (p 1/n , c) = . Thus, (γ) = ≥ (γ, n). f γ ≥0 U n n0 ∈ − ∩O− C Take any n n so that Leb( (γ, n)) > 0 and choose a point q (c 1/n, p ) ( ). 0 − γ f f By the maximality of J , the interval (c 1/n, p ) is not a wandering one. Thus, either − γ γ − ∩O− C (cid:54) (c 1/n, p ) is contained in the basin of a periodic-like attractor, or (c 1/n, p ) ( ) = ∅ γ γ f f . The ﬁrst situation is impossible, as it would imply that p would be attracted by γ the periodic-like attractor and then J wouldn’t be a wandering interval. So, we have − ∩ O− C (cid:54) ∅ γ (c 1/n, p ) ( ) = and we are free to choose q as above. γ f f (cid:40) \ C ∪ { } → Taking I = (q, c), we shall now consider two functions, g : [0, 1] ( q ) [0, 1], f j = 1, 2, given by ∈ f(x) if x / I g(x) = − ∈ , p + σ(f(x) p ) if x I γ γ | (cid:48)| − where σ = (2 sup f ) 1. | In Figure 6 we can compare the maps f and g. Notice that g is a contraction and ∀ ∈ ∈I \ ∪ ω (x) = p x I. Applying Lemma 15, we get for almost all x [0, 1] (B (g) B (g)) g γ (cid:91) 0 1 that ⊂ O ω (x) Re( +(α)), (11) g g ∈ \ α Cg Cg(I) where (cid:124) (cid:123)(cid:122) (cid:125) { ∈ O ∩ (cid:54) ∅} ⊃ { ∈ O ∩ (cid:54) ∅} ∪{ } (cid:51) C (I) = η C ; Re( +(η)) (q, c) = η C ; Re( +(η)) (q, c) = q+i γ. g g g f f C (I) f \ { − }∪ \ O O ∈ \ As C C ((cid:91)I) = ( q i C ) C ((cid:91)I) and as Re( +(η)) = Re( +(η)) for all η C C (I), −g ∈g f g g f f g if q i C (I) then g O O \ ⊂ \ { } Re( +(α)) = Re( +(α)) = A(C C (I)) A(C γ ) g f f g f ∈ \ ∈ \ α Cg Cg(I) α Cf Cg(I) ##LTLine## ##LTLine##

text: P.

text: BRANDA˜O,

text: J.

text: PALIS,

text: AND

text: V.

text: PINHEIRO

text: O

text: Given

text: K

text: C

text: let

text: A(K)

text: Re(

text: Thus,

text: Lemma

text: can

text: be

text: rewritten

text: as:

text: f

text: α

text: K

text: f

text: ω

text: (x)

text: for

text: almost

text: all

text: x

text: (B

text: (f)

text: B

text: f

text: f

text: Given

text: x

text: let

text: [C

text: α

text: C

text: ω

text: (x)

text: Notice

text: that

text: [C

text: [C

text: f

text: x

text: f

text: f

text: f

text: x

text: f

text: f

text: x

text: Lemma

text: (Parallax

text: II).

text: For

text: almost

text: all

text: x

text: B(f),

text: we

text: have

text: ω

text: (x)

text: A(C

text: γ

text: for

text: all

text: f

text: f

text: γ

text: C

text: [C

text: f

text: f

text: x

text: U

text: U

text: Proof.

text: Let

text: γ

text: C

text: and

text: set

text: (γ)

text: x

text: B(f)

text: γ

text: ω

text: (x)

text: that

text: is,

text: (γ)

text: x

text: f

text: f

text: U

text: B(f)

text: γ

text: C

text: [C

text: To

text: prove

text: the

text: claim

text: above,

text: we

text: may

text: assume

text: Leb(

text: (γ))

text: f

text: f

text: x

text: C

text: By

text: Lemma

text: that

text: γ

text: We

text: can

text: also

text: assume

text: that

text: γ

text: c

text: i

text: for

text: some

text: c

text: the

text: case

text: f

text: C

text: f

text: c

text: i

text: is

text: analogous.

text: As

text: γ

text: C

text: either

text: p

text: c

text: or

text: p

text: γ.

text: U

text: U

text: f

text: O

text: ∩f

text: f−

text: γ

text: ∅}γ

text: Let

text: (γ,

text: n)

text: x

text: (γ)

text: +(x)

text: (p

text: c)

text: and

text: let

text: n

text: be

text: such

text: that

text: C

text: U

text: f

text: γ

text: U

text: (p

text: c)

text: Thus,

text: (γ)

text: (γ,

text: n).

text: f

text: γ

text: ≥0

text: U

text: n

text: ∩O−

text: C

text: Take

text: any

text: n

text: n

text: so

text: that

text: Leb(

text: (γ,

text: n))

text: and

text: choose

text: a

text: point

text: q

text: (c

text: p

text: γ

text: f

text: f

text: By

text: the

text: maximality

text: of

text: J

text: the

text: interval

text: (c

text: p

text: is

text: not

text: a

text: wandering

text: one.

text: Thus,

text: either

text: γ

text: γ

text: ∩O−

text: C

text: (c

text: p

text: is

text: contained

text: in

text: the

text: basin

text: of

text: a

text: periodic-like

text: attractor,

text: or

text: (c

text: p

text: γ

text: γ

text: f

text: f

text: The

text: ﬁrst

text: situation

text: is

text: impossible,

text: as

text: it

text: would

text: imply

text: that

text: p

text: would

text: be

text: attracted

text: by

text: γ

text: the

text: periodic-like

text: attractor

text: and

text: then

text: J

text: wouldn’t

text: be

text: a

text: wandering

text: interval.

text: So,

text: we

text: have

text: O−

text: C

text: γ

text: (c

text: p

text: and

text: we

text: are

text: free

text: to

text: choose

text: q

text: as

text: above.

text: γ

text: f

text: f

text: C

text: Taking

text: I

text: (q,

text: c),

text: we

text: shall

text: now

text: consider

text: two

text: functions,

text: g

text: q

text: f

text: j

text: given

text: by

text: f(x)

text: if

text: x

text: I

text: g(x)

text: p

text: σ(f(x)

text: p

text: if

text: x

text: I

text: γ

text: γ

text: where

text: σ

text: sup

text: f

text: In

text: Figure

text: we

text: can

text: compare

text: the

text: maps

text: f

text: and

text: g.

text: Notice

text: that

text: g

text: is

text: a

text: contraction

text: and

text: ∈I

text: ω

text: (x)

text: p

text: x

text: I.

text: Applying

text: Lemma

text: we

text: get

text: for

text: almost

text: all

text: x

text: (B

text: (g)

text: B

text: (g))

text: g

text: γ

text: that

text: O

text: ω

text: (x)

text: Re(

text: +(α)),

text: g

text: g

text: α

text: Cg

text: Cg(I)

text: where

text: O

text: O

text: C

text: (I)

text: η

text: C

text: Re(

text: +(η))

text: (q,

text: c)

text: η

text: C

text: Re(

text: +(η))

text: (q,

text: c)

text: q+i

text: γ.

text: g

text: g

text: g

text: f

text: f

text: C

text: (I)

text: f

text: O

text: O

text: As

text: C

text: C

text: q

text: i

text: C

text: C

text: and

text: as

text: Re(

text: +(η))

text: Re(

text: +(η))

text: for

text: all

text: η

text: C

text: C

text: (I),

text: −g

text: ∈g

text: f

text: g

text: g

text: f

text: f

text: g

text: if

text: q

text: i

text: C

text: (I)

text: then

text: g

text: O

text: O

text: Re(

text: +(α))

text: Re(

text: +(α))

text: A(C

text: C

text: (I))

text: A(C

text: γ

text: g

text: f

text: f

text: g

text: f

text: α

text: Cg

text: Cg(I)

text: α

text: Cf

text: Cg(I)

text: ##LTLine##

text: ##LTLine##


text: (cid:9) 32 (cid:0) (cid:1) P. BRANDA˜O, J. PALIS, AND V. PINHEIRO \ \ O ∪ O { ∈ U ∩ (cid:54) ∅} ([0, 1] J) ( +(a) +(b)) such that Leb( x ; ω+(x) I ) = ) > 0 and that ∩ O Vf ∅ f g t0 ⊃ I Re ( +( ) = . Thus, by the interval dichotomy lemma (Proposition 11), ω (x) t0 g g ∈ ≥ g | I for almost every x I . By the homterval lemma, we get n 0 such that gn is t0 C ∩t0 { } ∩ (cid:54) ∅ ∩ (cid:54) ∅ It0 a diﬀeomorphism and gn(I ) = a, b, c gn(I ) = and so, gn(I ) J = . As a g t0 t0 t0 U ∩ consequence, the orbit (with respect to g and also to f) of almost every point of I U (cid:3)t0 intersects J, contradicting the deﬁnition of . \ Proposition 27 (Periodic-like attractors for contracting Lorenz maps). Let f : [0, 1] { } → c [0, 1] be a C3 contracting Lorenz map with negative Schwarzian derivative (f may be ﬂat). If f has a periodic-like attractor A , then either Leb(β (A )) = 1 or else there is 1 ∪ f 1 a second periodic-like attractor A such that Leb(β (A ) β (A )) = 1. 2 f 1 f 2 Proof. Let A be a periodic-like attractor for f. By Singer’s Theorem [38], there is ε > 0 −1 ⊂ ⊂ such that (c ε, c) or (c, c + ε) β (A ). Suppose for instance that (c, c + ε) β (A ), f 1 f 1 the other case being similar. Let (c, b) be the maximal open interval containing (c, c + ε) and contained in β (A ). f 1∃ − Firstly suppose that δ > 0 such that (c δ, c) belongs to the basin of attraction of a periodic-like attractor A (A may be equal to A ). We then apply Lemma 26 and conclude 2 2 1 the proof. − Thus, we may assume that, for every δ > 0, (c δ, c) is not contained in the basin of attraction of a periodic attractor. In particular, this implies, by Singer’s Theorem, that A is the unique periodic-like attractor of f. 1 Claim. If (a, c) is a homterval for some a < c, then Leb(β (A )) = 1. f 1 (cid:48) Proof of the claim. Let I = (a , c) be the maximal homterval containing (a, c). As I is a ∀ ∈ ∩ (cid:54) ∅ ⊂ homterval, ω (x) = ω (y), x, y I. Thus, if I β (A ) = then I β (A ) and by f f f 1 f 1 Lemma 26 we get that Leb(β (A )) = 1. f 1 ∩ ∅ Now, let us show that the assumption I β (A ) = leads to a contradiction. Indeed, ∩ ∅ ∩ ∅ ∀f 1≥ if I β (A ) = then fn(J) (c, b) = n 0. By the maximality of I, either (cid:48) f 1 (cid:48) ≥ ∩ ∅ ∀ ≥ a = 0 or f(cid:96)(a ) = c for some (cid:96) 1. As fn(J) (c, b) = n 0, we necessarily (cid:48) | ∈ − have a = 0. As f2 is a diﬀeomorphism, c / f(I) = (0, lim ↓ f(c ε)). This implies that ⊂ I ε 0 f(I) I and therefore, (0, c) is contained in the basin of attraction of some ﬁxed-like point ∈ p [0, c], and as we are assuming that A is the unique periodic-like attractor, A = p and ⊂ 1 1 I = (0, c) β (p), contradicting the assumption. f (cid:3) \ Let us suppose, by contradiction, that Leb([0, 1] β (A )) > 0. Thus, it follows from the f 1 claim above that f can not have an Ewi attractor. Also, by Lemma 4, f does not admit a cycle of intervals. Thus, it follows from Corollary 19 that − ∈ ∈ \ c i ω (x) for almost all x [0, 1] β (A ). (22) f f 1

text: P.

text: BRANDA˜O,

text: J.

text: PALIS,

text: AND

text: V.

text: PINHEIRO

text: O

text: O

text: U

text: J)

text: +(a)

text: +(b))

text: such

text: that

text: Leb(

text: x

text: ω+(x)

text: I

text: and

text: that

text: O

text: Vf

text: f

text: g

text: I

text: Re

text: Thus,

text: by

text: the

text: interval

text: dichotomy

text: lemma

text: (Proposition

text: ω

text: (x)

text: g

text: g

text: g

text: I

text: for

text: almost

text: every

text: x

text: I

text: By

text: the

text: homterval

text: lemma,

text: we

text: get

text: n

text: such

text: that

text: gn

text: is

text: C

text: a

text: diﬀeomorphism

text: and

text: gn(I

text: a,

text: b,

text: c

text: gn(I

text: and

text: so,

text: gn(I

text: J

text: As

text: a

text: g

text: U

text: consequence,

text: the

text: orbit

text: (with

text: respect

text: to

text: g

text: and

text: also

text: to

text: f)

text: of

text: almost

text: every

text: point

text: of

text: I

text: U

text: intersects

text: J,

text: contradicting

text: the

text: deﬁnition

text: of

text: Proposition

text: (Periodic-like

text: attractors

text: for

text: contracting

text: Lorenz

text: maps).

text: Let

text: f

text: c

text: be

text: a

text: contracting

text: Lorenz

text: map

text: with

text: negative

text: Schwarzian

text: derivative

text: (f

text: may

text: be

text: ﬂat).

text: If

text: f

text: has

text: a

text: periodic-like

text: attractor

text: A

text: then

text: either

text: Leb(β

text: (A

text: or

text: else

text: there

text: is

text: f

text: a

text: second

text: periodic-like

text: attractor

text: A

text: such

text: that

text: Leb(β

text: (A

text: β

text: (A

text: f

text: f

text: Proof.

text: Let

text: A

text: be

text: a

text: periodic-like

text: attractor

text: for

text: f.

text: By

text: Singer’s

text: Theorem

text: there

text: is

text: ε

text: −1

text: such

text: that

text: (c

text: ε,

text: c)

text: or

text: (c,

text: c

text: ε)

text: β

text: (A

text: Suppose

text: for

text: instance

text: that

text: (c,

text: c

text: ε)

text: β

text: (A

text: f

text: f

text: the

text: other

text: case

text: being

text: similar.

text: Let

text: (c,

text: b)

text: be

text: the

text: maximal

text: open

text: interval

text: containing

text: (c,

text: c

text: ε)

text: and

text: contained

text: in

text: β

text: (A

text: f

text: 1∃

text: Firstly

text: suppose

text: that

text: δ

text: such

text: that

text: (c

text: δ,

text: c)

text: belongs

text: to

text: the

text: basin

text: of

text: attraction

text: of

text: a

text: periodic-like

text: attractor

text: A

text: (A

text: may

text: be

text: equal

text: to

text: A

text: We

text: then

text: apply

text: Lemma

text: and

text: conclude

text: the

text: proof.

text: Thus,

text: we

text: may

text: assume

text: that,

text: for

text: every

text: δ

text: (c

text: δ,

text: c)

text: is

text: not

text: contained

text: in

text: the

text: basin

text: of

text: attraction

text: of

text: a

text: periodic

text: attractor.

text: In

text: particular,

text: this

text: implies,

text: by

text: Singer’s

text: Theorem,

text: that

text: A

text: is

text: the

text: unique

text: periodic-like

text: attractor

text: of

text: f.

text: Claim.

text: If

text: (a,

text: c)

text: is

text: a

text: homterval

text: for

text: some

text: a

text: c,

text: then

text: Leb(β

text: (A

text: f

text: Proof

text: of

text: the

text: claim.

text: Let

text: I

text: (a

text: c)

text: be

text: the

text: maximal

text: homterval

text: containing

text: (a,

text: c).

text: As

text: I

text: is

text: a

text: homterval,

text: ω

text: (x)

text: ω

text: (y),

text: x,

text: y

text: I.

text: Thus,

text: if

text: I

text: β

text: (A

text: then

text: I

text: β

text: (A

text: and

text: by

text: f

text: f

text: f

text: f

text: Lemma

text: we

text: get

text: that

text: Leb(β

text: (A

text: f

text: Now,

text: let

text: us

text: show

text: that

text: the

text: assumption

text: I

text: β

text: (A

text: leads

text: to

text: a

text: contradiction.

text: Indeed,

text: ∀f

text: 1≥

text: if

text: I

text: β

text: (A

text: then

text: fn(J)

text: (c,

text: b)

text: n

text: By

text: the

text: maximality

text: of

text: I,

text: either

text: f

text: a

text: or

text: c

text: for

text: some

text: As

text: fn(J)

text: (c,

text: b)

text: n

text: we

text: necessarily

text: have

text: a

text: As

text: is

text: a

text: diﬀeomorphism,

text: c

text: f(I)

text: lim

text: f(c

text: ε)).

text: This

text: implies

text: that

text: I

text: ε

text: f(I)

text: I

text: and

text: therefore,

text: c)

text: is

text: contained

text: in

text: the

text: basin

text: of

text: attraction

text: of

text: some

text: ﬁxed-like

text: point

text: p

text: c],

text: and

text: as

text: we

text: are

text: assuming

text: that

text: A

text: is

text: the

text: unique

text: periodic-like

text: attractor,

text: A

text: p

text: and

text: I

text: c)

text: β

text: (p),

text: contradicting

text: the

text: assumption.

text: f

text: Let

text: us

text: suppose,

text: by

text: contradiction,

text: that

text: β

text: (A

text: Thus,

text: it

text: follows

text: from

text: the

text: f

text: claim

text: above

text: that

text: f

text: can

text: not

text: have

text: an

text: Ewi

text: attractor.

text: Also,

text: by

text: Lemma

text: f

text: does

text: not

text: admit

text: a

text: cycle

text: of

text: intervals.

text: Thus,

text: it

text: follows

text: from

text: Corollary

text: that

text: c

text: i

text: ω

text: (x)

text: for

text: almost

text: all

text: x

text: β

text: (A

text: f

text: f


text: 34 P. BRANDA˜O, J. PALIS, AND V. PINHEIRO (cid:0) (cid:84) (cid:1) ∀ ∈ ∀ ∈ ⊂ P Because Gn(x) = F n(x) x V (p) and n N, we get G(V (p)) V (p). Let be U ∀ ∈ P the collection of connected components of . As, G(P ) = (p, q) P , SG < 0 and − U ≥ Leb ≥ G n( ) Leb(V (p)) > 0, it follows from Lemma 10 that ω (x) = ω (x) = n 0 ∈ F ∈ G [p, q] for almost every x V (p). In particular, the ω-limit set of almost every x V (p) is a cycle of intervals. This is a contradiction, as f can not admit a cycle of intervals. Thus, \ we necessarily have Leb([0, 1] β (A )) = 0, which concludes the proof. f 1 (cid:3) If f has attracting periodic-like orbits, then the proof follows from Proposition 27 above. If f does not admit attracting periodic-like orbits, then it follows from Corollary 25 that ∈ ω (x) = A for almost every x [0, 1], where f O − ∪ O O ∪ O A = Re( +(c i)) Re( +(c + i)) = +(c−) +(c ) f f f f + O { ± ≥ } O − ∪ and +(c±) = lim ↓ fn(c ε) ; n 0 . Thus, we need only to verify that Re( +(c i)) O f ε 0 f Re( +(c + i)) is a perfect set and this follows straightforwardly from Corollary 25 and f Corollary 13. In the present paper we have dealt with metrical attractors. In this kind of attractor, the basin of attraction is required to have positive Lebesgue measure. Another viewpoint is to consider topological attractors, namely the ones whose basins of attraction are residual in some open set of the domain, see Milnor [28]. Notice that the topological attractors and the metrical ones may not be the same, as in the case of wild attractors [8]. For non-ﬂat C3 maps of the interval [0, 1] with negative Schwarzian derivative it is known that the number of topological attractors is bounded by the number of critical points, as shown by Guckenheimer [14] and Blokh and Lyubich [6]. For maps with discontinuities, Branda˜o showed in [4] the topological version of Theorem C, that is, contracting Lorenz maps have either one single topological attractor or two attracting periodic orbits whose union of basins of attraction is a residual subset of the whole interval. In the context of topological attractors, the question of ﬁniteness of the number of attractors for discontin- uous maps with more than one critical point remains open. References [1] L. Alsed`a, J. Llibre, M. Misiurewicz. Periods and entropy for Lorenz-like maps, Ann. Inst. Fourier, 1989. [2] A. Arneodo, P. Coullet, C. Tresser. A possible new mechanism for the onset of turbulence, Phys. Lett. A 81(4), 197-201, 1981. [3] A. Andronov, L. Pontrjagin. Syst`emes Grossiers. Dokl. Akad. Nauk., SSSR, 14, 247-251, 1937. [4] P. Brand˜ao. On the Structure of Contracting Lorenz maps, preprint, IMPA Thesis, 2013. [5] A. Blokh, M. Lyubich. Ergodicity of transitive maps of the interval, Ukrainian Math. J. 41, 985–988, 1989. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P.

text: BRANDA˜O,

text: J.

text: PALIS,

text: AND

text: V.

text: PINHEIRO

text: P

text: Because

text: Gn(x)

text: F

text: n(x)

text: x

text: V

text: (p)

text: and

text: n

text: N,

text: we

text: get

text: G(V

text: (p))

text: V

text: (p).

text: Let

text: be

text: U

text: P

text: the

text: collection

text: of

text: connected

text: components

text: of

text: As,

text: G(P

text: (p,

text: q)

text: P

text: SG

text: and

text: U

text: Leb

text: G

text: n(

text: Leb(V

text: (p))

text: it

text: follows

text: from

text: Lemma

text: that

text: ω

text: (x)

text: ω

text: (x)

text: n

text: F

text: G

text: [p,

text: q]

text: for

text: almost

text: every

text: x

text: V

text: (p).

text: In

text: particular,

text: the

text: ω-limit

text: set

text: of

text: almost

text: every

text: x

text: V

text: (p)

text: is

text: a

text: cycle

text: of

text: intervals.

text: This

text: is

text: a

text: contradiction,

text: as

text: f

text: can

text: not

text: admit

text: a

text: cycle

text: of

text: intervals.

text: Thus,

text: we

text: necessarily

text: have

text: β

text: (A

text: which

text: concludes

text: the

text: proof.

text: f

text: If

text: f

text: has

text: attracting

text: periodic-like

text: orbits,

text: then

text: the

text: proof

text: follows

text: from

text: Proposition

text: above.

text: If

text: f

text: does

text: not

text: admit

text: attracting

text: periodic-like

text: orbits,

text: then

text: it

text: follows

text: from

text: Corollary

text: that

text: ω

text: (x)

text: A

text: for

text: almost

text: every

text: x

text: where

text: f

text: O

text: O

text: O

text: O

text: A

text: Re(

text: +(c

text: i))

text: Re(

text: +(c

text: i))

text: +(c−)

text: +(c

text: f

text: f

text: f

text: f

text: O

text: O

text: and

text: +(c±)

text: lim

text: fn(c

text: ε)

text: n

text: Thus,

text: we

text: need

text: only

text: to

text: verify

text: that

text: Re(

text: +(c

text: i))

text: O

text: f

text: ε

text: f

text: Re(

text: +(c

text: i))

text: is

text: a

text: perfect

text: set

text: and

text: this

text: follows

text: straightforwardly

text: from

text: Corollary

text: and

text: f

text: Corollary

text: In

text: the

text: present

text: paper

text: we

text: have

text: dealt

text: with

text: metrical

text: attractors.

text: In

text: this

text: kind

text: of

text: attractor,

text: the

text: basin

text: of

text: attraction

text: is

text: required

text: to

text: have

text: positive

text: Lebesgue

text: measure.

text: Another

text: viewpoint

text: is

text: to

text: consider

text: topological

text: attractors,

text: namely

text: the

text: ones

text: whose

text: basins

text: of

text: attraction

text: are

text: residual

text: in

text: some

text: open

text: set

text: of

text: the

text: domain,

text: see

text: Milnor

text: Notice

text: that

text: the

text: topological

text: attractors

text: and

text: the

text: metrical

text: ones

text: may

text: not

text: be

text: the

text: same,

text: as

text: in

text: the

text: case

text: of

text: wild

text: attractors

text: For

text: non-ﬂat

text: maps

text: of

text: the

text: interval

text: with

text: negative

text: Schwarzian

text: derivative

text: it

text: is

text: known

text: that

text: the

text: number

text: of

text: topological

text: attractors

text: is

text: bounded

text: by

text: the

text: number

text: of

text: critical

text: points,

text: as

text: shown

text: by

text: Guckenheimer

text: and

text: Blokh

text: and

text: Lyubich

text: For

text: maps

text: with

text: discontinuities,

text: Branda˜o

text: showed

text: in

text: the

text: topological

text: version

text: of

text: Theorem

text: C,

text: that

text: is,

text: contracting

text: Lorenz

text: maps

text: have

text: either

text: one

text: single

text: topological

text: attractor

text: or

text: two

text: attracting

text: periodic

text: orbits

text: whose

text: union

text: of

text: basins

text: of

text: attraction

text: is

text: a

text: residual

text: subset

text: of

text: the

text: whole

text: interval.

text: In

text: the

text: context

text: of

text: topological

text: attractors,

text: the

text: question

text: of

text: ﬁniteness

text: of

text: the

text: number

text: of

text: attractors

text: for

text: discontin-

text: uous

text: maps

text: with

text: more

text: than

text: one

text: critical

text: point

text: remains

text: open.

text: References

text: L.

text: Alsed`a,

text: J.

text: Llibre,

text: M.

text: Misiurewicz.

text: Periods

text: and

text: entropy

text: for

text: Lorenz-like

text: maps,

text: Ann.

text: Inst.

text: Fourier,

text: A.

text: Arneodo,

text: P.

text: Coullet,

text: C.

text: Tresser.

text: A

text: possible

text: new

text: mechanism

text: for

text: the

text: onset

text: of

text: turbulence,

text: Phys.

text: Lett.

text: A

text: A.

text: Andronov,

text: L.

text: Pontrjagin.

text: Syst`emes

text: Grossiers.

text: Dokl.

text: Akad.

text: Nauk.,

text: SSSR,

text: P.

text: Brand˜ao.

text: On

text: the

text: Structure

text: of

text: Contracting

text: Lorenz

text: maps,

text: preprint,

text: IMPA

text: Thesis,

text: A.

text: Blokh,

text: M.

text: Lyubich.

text: Ergodicity

text: of

text: transitive

text: maps

text: of

text: the

text: interval,

text: Ukrainian

text: Math.

text: J.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:12) (cid:12) 10 (cid:12) P. BR(cid:12)ANDA˜O, J. PALIS, AND V. PINHEIRO (cid:12) (cid:12) (cid:12) (cid:12) (cid:48) (F n) (p) ≤ ∀ ∈ ∀ ∀ ∈ − (F n)(cid:48)(q) K, p, q Jn(x), n, x F n(J), (2) − where J (x) is the connected component of F n(J) that contains x. n∈ Given x I let U { ∈ ∈ − } (x) = n N ; x F n(J) . | | ∈ (cid:48) Claim. We have limU (cid:51) →∞ J (x) = 0, for Lebesgue almost every x V . (x) n n (cid:48) Proof of the claim. Suppose that there is a Lebesgue density point x of V such that (cid:84) | | lim J (x) > 0. U (cid:51) →∞ n (x) n ⊃ · · · ⊃ (cid:38) Then, M := Interior( ∈U J (x)) is an open interval with J (x) J (x) M. As (cid:83)∀ n (x) n 1 n −→→∞ F n(J (x)(cid:102)) = J n, it follows from the bounded distortion (2) above that F(cid:102)n(M)(cid:102)n J. n ∩ (cid:54) ∅ | ∀ Thus, there exists (cid:96) big enough so that F (cid:96)(M|(cid:102)) M = . As F n(cid:96) M is diﬀeomorphism n, deﬁning M = ≥ F n(cid:102)(cid:96)(M) we h(cid:102)ave that F (cid:96) is a diﬀeomorphism and F (cid:96)(M) = M. This n 0 | M implies, since SF < 0, that F (cid:96) has one, p , or at most two attracting ﬁxed-like points \ M 1 { } { } p , p such that Leb(M β(A)) = 0, where A = p or p , p . Therefore, Lebesgue 1 2 1 1 2 almost every point of the neighborhood of x is contained in the basin of attraction of some (cid:48) attracting periodic-like orbit, contradicting the fact that V does not intersect the basin of (cid:48) (cid:3) any periodic-like attractor (recall that x is a density point of V ). ⊃ (cid:48) ∩ (cid:48) (cid:48) It follows from the claim above that Per(F ) V (a , b ). Furthermore, using the claim, (cid:48) (cid:48) (cid:48) (cid:0)\ (cid:48) the invariance of V and the bounded distortion (2), we also get that Leb((a , b ) V ) = 0. (cid:1) ⊃ (cid:48) (cid:48) As a consequence, Per(F ) (a , b ). ∈ (cid:48) (cid:48) ∩ (cid:0){ ∈ (cid:48) ∩(cid:1) O ∪ Let α , β (a , b ) Per(f) be such that α < β , Leb( x V ; ω (x) +(α ) O 0(cid:54) 0∅} { ∈ (cid:48) ∩ 0 0 (cid:54) ∅} F f 0 +(β ) = ) = 0 and Leb( x V , ω (x) (α , β ) = ) > 0. f 0 F 0 0 \ O ∪ O Let T = (α, β) be a connected component of (α , β ) +(α ) +(β ) such that { ∈ (cid:48) ∩ (cid:54) ∅} ∗ { 0∈ 0 O F 0 ∩ F(cid:54) 0∅} T Leb( x V ; ω (x) T = ) > 0. Let T = x T ; +(F (x)) T = , be the F (cid:12) (cid:12) ∗ ∗ → F sOet of all∩connec∅ted com(cid:12)ponents o(cid:12) f T and FT : T ∈T Tbe the ﬁrst return map to T . As F+(∂T ) T = , we ha(cid:12)(cid:12)ve that F(cid:12)(cid:12)T (W ) = T for all W . It follows again from (2) that (cid:48) (F n) (p) ≤ ∀ ∈ T ∀ ∀ ∈ (cid:48)(cid:48) (FTn)(cid:48)(q) K, p, q n(x), n, x V , T (cid:48)(cid:48) { ∈ ∩ (cid:48) ∩ (cid:54) ∅} T where V = x T V ; ω (x) T = and (x) is the connected component of − F n \ (cid:48) F n(T ) containing x. That is, F satisﬁes the hypothesis of Lemma 9. Thus, Leb(T V ) = T \ (cid:48)(cid:48) ⊃ T ∈ ∈ (cid:48)∩ Leb(T V ) = 0 and ω (x) [α, β] for Lebesgue almost every x T . Chose any p V T ≥ FT ⊂ (cid:48) (cid:48) ◦ − (cid:28) and n 1 big enough so that J (p) T . As F n(J (p)) = (a , b ) and Leb F 1 Leb, (cid:48) (cid:48) \ n (cid:48) (cid:48) \ (cid:48) n ⊃ (cid:48) (cid:48) we have that Leb((a , b ) V ) = Leb((a , b ) V ) = 0 and ω (x) [a , b ] for almost all ∈ (cid:48) (cid:48) (cid:48) (cid:48) F x (a , b ). Finally, as we can take a as close to a and b as close to b as wished, we \ ∈ conclude that Leb(I V ) = 0 and ω (x) = [a, b] for Lebesgue almost every x [a, b]. F (cid:3) ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P.

text: J.

text: PALIS,

text: AND

text: V.

text: PINHEIRO

text: (F

text: n)

text: (p)

text: (F

text: K,

text: p,

text: q

text: Jn(x),

text: n,

text: x

text: F

text: n(J),

text: where

text: J

text: (x)

text: is

text: the

text: connected

text: component

text: of

text: F

text: n(J)

text: that

text: contains

text: x.

text: n∈

text: Given

text: x

text: I

text: let

text: U

text: (x)

text: n

text: N

text: x

text: F

text: n(J)

text: Claim.

text: We

text: have

text: limU

text: J

text: (x)

text: for

text: Lebesgue

text: almost

text: every

text: x

text: V

text: (x)

text: n

text: n

text: Proof

text: of

text: the

text: claim.

text: Suppose

text: that

text: there

text: is

text: a

text: Lebesgue

text: density

text: point

text: x

text: of

text: V

text: such

text: that

text: lim

text: J

text: (x)

text: U

text: n

text: (x)

text: n

text: Then,

text: M

text: Interior(

text: ∈U

text: J

text: (x))

text: is

text: an

text: open

text: interval

text: with

text: J

text: (x)

text: J

text: (x)

text: M.

text: As

text: n

text: (x)

text: n

text: n

text: F

text: n(J

text: J

text: n,

text: it

text: follows

text: from

text: the

text: bounded

text: distortion

text: above

text: that

text: J.

text: n

text: Thus,

text: there

text: exists

text: big

text: enough

text: so

text: that

text: F

text: M

text: As

text: F

text: M

text: is

text: diﬀeomorphism

text: n,

text: deﬁning

text: M

text: F

text: we

text: that

text: F

text: is

text: a

text: diﬀeomorphism

text: and

text: F

text: M.

text: This

text: n

text: M

text: implies,

text: since

text: SF

text: that

text: F

text: has

text: one,

text: p

text: or

text: at

text: most

text: two

text: attracting

text: ﬁxed-like

text: points

text: M

text: p

text: p

text: such

text: that

text: Leb(M

text: β(A))

text: where

text: A

text: p

text: or

text: p

text: p

text: Therefore,

text: Lebesgue

text: almost

text: every

text: point

text: of

text: the

text: neighborhood

text: of

text: x

text: is

text: contained

text: in

text: the

text: basin

text: of

text: attraction

text: of

text: some

text: attracting

text: periodic-like

text: orbit,

text: contradicting

text: the

text: fact

text: that

text: V

text: does

text: not

text: intersect

text: the

text: basin

text: of

text: any

text: periodic-like

text: attractor

text: (recall

text: that

text: x

text: is

text: a

text: density

text: point

text: of

text: V

text: It

text: follows

text: from

text: the

text: claim

text: above

text: that

text: Per(F

text: V

text: (a

text: b

text: Furthermore,

text: using

text: the

text: claim,

text: the

text: invariance

text: of

text: V

text: and

text: the

text: bounded

text: distortion

text: we

text: also

text: get

text: that

text: Leb((a

text: b

text: V

text: As

text: a

text: consequence,

text: Per(F

text: (a

text: b

text: O

text: Let

text: α

text: β

text: (a

text: b

text: Per(f)

text: be

text: such

text: that

text: α

text: β

text: Leb(

text: x

text: V

text: ω

text: (x)

text: +(α

text: O

text: F

text: f

text: +(β

text: and

text: Leb(

text: x

text: V

text: ω

text: (x)

text: (α

text: β

text: f

text: F

text: O

text: O

text: Let

text: T

text: (α,

text: β)

text: be

text: a

text: connected

text: component

text: of

text: (α

text: β

text: +(α

text: +(β

text: such

text: that

text: 0∈

text: O

text: F

text: T

text: Leb(

text: x

text: V

text: ω

text: (x)

text: T

text: Let

text: T

text: x

text: T

text: +(F

text: (x))

text: T

text: be

text: the

text: F

text: F

text: sOet

text: of

text: all∩connec∅ted

text: f

text: T

text: and

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