TThe Scenery Flow for Self-Affine Measures.
Tom KemptonApril 18, 2019
Abstract
We describe the scaling scenery associated to Bernoulli measures supported onseparated self-affine sets under the condition that certain projections of the mea-sure are absolutely continuous.
The scenery flow is an extremely useful tool for studying fractal sets and measures.Recently several long standing conjectures in fractal geometry have been resolvedusing the scenery flow. In particular, Furstenberg proved a dimension conservationresult for uniformly scaling measures which generate ergodic fractal distributionsand Hochman and Shmerkin gave conditions under which every projection of afracal measure µ has dimension equal to min { dim H ( µ ) , } , [13, 15]. The sceneryflow has also been used to prove several important results in geometric measuretheory, [18, 22, 25]. For this reason, much attention has been given recently to theproblem of understanding the scenery flow for various classes of fractal measures,and in particular the question of whether they are uniformly scaling and whetherthey generate ergodic fractal distributions.The scenery flow for non-overlapping self-similar and self-conformal measures iswell understood, [4, 5, 12, 23]. In the self-affine setting, the scenery flow haspreviously been studied for measures on Bedford-McMullen carpets, [11, 1], andHochman asked whether it can be understood more generally [14]. In this articlewe study the scenery flow for a wide class of self-affine measures which satisfy acone condition and a projection condition, given later.There is much interesting dynamics associated to self-affine sets and measureswhich is not present in the self-similar case. In particular, iterated function sys-tems defining a self-affine set give rise to further iterated function systems onprojective space which describe the way in which straight lines through the originare mapped onto each other by affine maps. This second iterated function systemdefines the Furstenberg measure on projective space, which is crucial to under-standing self-affine measures. Recently formulae for the Hausdorff dimension of aself-affine set were given in terms of the dimension of projections of the self-affinemeasure in typical directions chosen according to the Furstenberg measure, [2, 8]. a r X i v : . [ m a t h . D S ] M a y imension theory for self-affine sets is an extremely active topic of research, seefor example the survey papers [7, 9], and yet a general theory does not yet exist.We hope that as the understanding of the scenery flow for self-affine sets becomesmore developed, a general theory of dimension for self-affine sets may emerge.In this article we build on our work with Falconer on the dynamics of self-affinesets, [8], to describe the scenery flow for self-affine measures associated to strictlypositive matrices under the condition that projections of the self-affine measurein typical directions for the Furstenberg measure are absolutely continuous. Thisprojection condition holds typically on large parts of parameter space, [3], andholds everywhere for some open sets in parameter space [8]. Very recently thescenery flow for self-affine sets rather than measures was studied in [17]. Addi-tionally, we study the scenery flow for slices through self-affine measures withoutassuming any condition on projections. Let M denote the space of Borel probability measures µ supported on the unitdisk X with 0 ∈ supp ( µ ). Let d denote the Prokhorov metric on M , given by d ( µ, ν ) := inf { (cid:15) : µ ( A ) ≤ ν ( A (cid:15) ) + (cid:15), ν ( A ) ≤ µ ( A (cid:15) ) + (cid:15) for all Borel sets A } where A (cid:15) := { x ∈ R : d ( x, y ) < (cid:15) for some y ∈ A } . The Prokhorov metricmetrises the weak ∗ topology.Let B ( x, r ) denote the ball of radius r centred at x ∈ R . Given µ ∈ M we let S t ( µ )denote the measure µ | B (0 ,e − t ) , normalised to have mass 1 and mapped onto theunit disk by the dilation map x → e t x for x ∈ R . Note that S t + s ( µ ) = S t ( S s ( µ ))and so S is a well defined flow on the space M .We refer to S t -invariant measures P on the space M as distributions. Apply-ing ergodic theory to the system ( M , P, S t ) turns out to be extremely useful ingeometric measure theory and the study of fractals.The flow S t describes the process of zooming in on the measure µ around theorigin. If we are interested in zooming in on some other point x ∈ R we canfirst apply the map T x given by T x ( y ) := y − x and then apply S t to the resultingmeasure. For shorthand, we let S t,x ( µ ) := S t ◦ T x ( µ ).We let < µ > T,x := 1 T (cid:90) T δ S t,x ( µ ) dt be called the scenery distribution of µ at x up to time T . < µ > T,x gives mass1 T (cid:90) T χ A ( S t,x ( µ )) dt to Borel subsets A of M . If < µ > T,x → P in the weak ∗ topology as t → ∞ we saythat µ generates P at x . The measure µ is known as a uniformly scaling measure if it generates the same distribution P at µ -almost every point x , in which casewe say µ generates P . he quasi-Palm property is a property of S t -invariant distributions which describesa kind of translation invariance. We say that a distribution P is quasi-Palm if, fora subset A of M we have P ( A ) = 0 if and only if the measures S ,x ( µ ) obtainedby choosing µ according to P , choosing x according to µ are almost surely not in A . See [14] for a more full discussion of the quasi-Palm property.An ergodic fractal distribution is an S t invariant, ergodic probability distributionon M which is quasi-Palm. The best case scenario for inferring properties ofmeasures µ from the distributions they generate is that µ is a uniformly scal-ing measure generating an ergodic fractal distribution, this will not be the casefor the class of self-affine measures which we consider because of a rotation ele-ment which depends upon the point around which we are zooming in, but if wereto disregard this rotational effect then the generated measures would indeed beuniformly scaling. We begin by studying an example of a self-affine measure which demonstratesthe extra difficulties associated with studying the scenery flow for self-affine, asopposed to self-similar, measures. This example also demonstrates how, whencertain relevant projections of the self-affine measure are absolutely continuous,these extra difficulties can be overcome.The examples we study are a class of self-affine carpets first studied by Przyty-cki and Urbanski [24]. These carpets have rather less structure than Bedford-McMullen carpets, and so previous techniques of [11, 1] cannot be applied. Forthis example we scale along squares rather than balls.For λ ∈ ( ,
1) consider the self affine set E λ ⊂ [0 , which is the attractor of theiterated function system given by contractions T ( x, y ) = (cid:16) λx, y (cid:17) , T ( x, y ) = (cid:18) λx + (1 − λ ) , y + 23 (cid:19) . x Figure 1: The first two levels of E . or a · · · a n ∈ { , } n we let E a ··· a n := T a ··· a n ( E λ )where T a ··· a n := T a ◦ T a ◦ · · · T a n . Each point ( x, y ) ∈ E λ has a unique code a ∈ { , } N such that ( x, y ) ∈ E a ··· a n ∀ n ∈ N . We define the map π : { , } N → E λ to be the map from a code to the corre-sponding point ( x, y ) ∈ E λ . Let µ be the measure which arises from mapping the( , ) Bernoulli measure on { , } N to E λ by the coding map π .Now given a point π ( a ) ∈ E λ we let B ( π ( a ) , − n ) denote the square, centredat π ( a ), of side length 2 . − n . We further denote R ( σ n ( a ) , (3 λ ) − n ) the rectanglecentred at π ( σ n ( a )) of height 2 and width 2 . (3 λ ) − n < S n log 3 ( µ, a ) obtained by taking µ | B ( π ( a ) , − n ) µ ( B ( π ( a ) , − n )) and linearly rescaling it to live on the square [ − , . If our maps T i were non-overlapping similarities, this rescaling would be a rather straightforward process,we would just need to apply inverses of our contraction T i which would scale upthe small square to get a large square.Since our maps T i are affine contractions but not similarities, we instead needto apply a two step process. Our square B ( π ( a ) , − n ) intersects precisely onelevel n rectangle in the construction of E , namely the rectangle E a ··· a n . Firstwe apply the map T − a ··· a n to B ( π ( a ) , − n ) to get the rectangle R ( σ n ( a ) , (3 λ ) − n ).Here σ denotes the shift map on { , } N . The self-affinity relation for µ gives that µ | R ( σ n ( a ) , (3 λ ) − n ) is an affine copy of the measure µ | B ( π ( a ) , − n ) .To complete our process, we need to stretch the rectangle R ( σ n ( a ) , (3 λ ) − n ) hor-izontally by a factor of (3 λ ) n and translate the resulting square onto [ − , .Denote by D ( b, n ) the map which stretches the rectangle R ( b, (3 λ ) − n ) linearlyonto [ − , . We have S t,a ( µ ) = (cid:18) µ | B ( π ( a ) , − n ) µ ( B ( π ( a ) , − n )) (cid:19) ◦ T a ··· a n ◦ D ( σ n ( a ) , n ) − (1)There are three key observations which allow us to understand the scenery flow forthis example, and for the broader class of self-affine measures considered below. Observation 1:
We have (cid:18) µ | B ( π ( a ) , − n ) µ ( B ( π ( a ) , − n )) (cid:19) ◦ T a ··· a n = µ | R ( σ n ( a ) , (3 λ ) − n ) µ ( R ( σ n ( a ) , (3 λ ) − n ))This follows directly from the self-affinity of the measure µ . Observation 2:
Suppose that, for µ -almost every b ∈ { , } N , the sequence ofmeasures µ | R ( b, (3 λ ) − n ) µ ( R ( b, (3 λ ) − n )) ◦ D ( b, n ) − n [ − , converges weak ∗ to some limit measure µ b as n → ∞ . Then for all (cid:15) > a ∈ { , } N there exists a set A ⊂ N withlim n →∞ n | A ∩ { , · · · , n }| > − (cid:15) such that the sequence of measures S n log 3 ,a ( µ ) restricted to n ∈ A is weakly-asymptotic to the sequence of measures µ σ n ( a ) . Hence by the ergodicity of thesystem ( { , } N , σ, µ ) we have that µ is a uniformly scaling measure generating anergodic fractal distribution.This follows immediately from equation 1. We use Egorov’s theorem to turnalmost everywhere convergence to µ b into uniform convergence on a large set of b ∈ { , } N , which in turn allows us to generate the set A . Observation 3:
Suppose that the projection of µ onto the horizontal axis isabsolutely continuous. Then the limit measures µ b of Observation 2 exist for µ -almost every b , and hence µ is a uniformly scaling measure generating an ergodicfractal distribution.Observation 3 is less straightforward than the previous two. It relies firstly onthe fact that one can disintegrate a measure by vertical slicing. Secondly we useLemma 2.1, given below, which says that the scenery flow converges ν -almosteverywhere for measures ν which are absolutely continuous. The measures µ b take the form of a vertical slice of µ through b crossed with Lebesgue measure.The slice measures were described in [19]. We do not give further justificationfor observation 3 here, the corresponding proposition applying to more generalself-affine measures is proved later.An important result on which we rely is the following version of the Lebesguedensity theorem. Lemma 2.1.
Given an absolutely continuous measure ν on [ − , , for ν -almostevery x the scenery flow applied to ν around x converges to Lebesgue measure. Note that the projected measures of Observation 3 are a well studied family of self-similar measures known as Bernoulli convolutions, which are absolutely continuousfor all λ ∈ ( ,
1) outside of a family of exceptions which has Hausdorff dimension0 [26]. Thus, combining observations 1,2 and 3, we have the following theorem.
Theorem 2.1.
For all λ ∈ ( , outside of a set of exceptions of Hausdorffdimension zero, the (cid:0) , (cid:1) -Bernoulli measures µ on the fractal E λ are uniformlyscaling measures which generate an ergodic fractal distribution. A proof of this theorem follows fairly directly from the above three observations.We prefer to regard it as a corollary to the more general Theorem 6.1.The fact that our results for this example hold only for measures on sets E λ for which the corresponding Bernoulli convolution is absolutely continuous mayseem like a significant restriction, essentially we are restricting to the case thatwe already understand quite well. However, as one generalises from the carpet ike case of this example to more general self-affine sets the absolute continuityof ‘relevant projections’ becomes rather more natural, and the theorems that weprove later can be shown to hold for open sets in parameter space. Putting together the three observations above allows one to describe the sceneryflow for the measure µ under the condition that image under vertical projection of µ is an absolutely continous measure. A similar projection condition is requiredin the later, more general situation.One might hope to be able to prove the same results about µ under the looser pro-jection condition that the vertical projection of µ is a uniformly scaling measuregenerating an ergodic fractal distribution, i.e. rather than requiring the conver-gence of the scenery flow for typical points in the projected measure, one wouldonly require that the scenery flow on the projected measure is asymptotic to anergodic flow.The issue here is that one would have to do consider two ergodic maps simulta-neously, the first map b → σ n ( b ) governing the way in which the centre point ofObservation 1 moves, and the second map doing the time n log(3 λ ) scenery flowon the vertical projection of µ around point π ( σ n ( b )). We are unable to guaranteethat there is no resonance between these two ergodic maps and that the result-ing flow generates the ergodic distributions expected. This may be fixable in thespecific example of this section, but in the more general setting which follows itappears out of reach for the moment. Let k ∈ N and for each i ∈ { , · · · k } let A i be a real valued 2 × A i is is strictlypositive, for discussion of this ‘cone condition’ and how it can be relaxed see thefinal section.For each i ∈ { , · · · k } let d i ∈ R and let T i : R → R be given by T i ( x ) := A i ( x ) + d i . We assume a very strong separation condition, that the maps T i map the unitdisk into disjoint ellipses contained within the unit disk. This separation conditioncan most likely be weakened somewhat, we do not pursue this here. While theexamples of section 2 do not fit directly into our setting, since the associatedmatrices are not strictly positive, by rotating R they can be made to fit in theabove setting. he attractor E of our iterated function system is the unique non-empty compactset satisfying E = k (cid:91) i =1 T i ( E ) . Let T a ··· a n := T a ◦ T a ◦ · · · T a n and E a ··· a n := T a ··· a n ( E )for a · · · a n ∈ { , } n . Let X denote the unit disk and let X a ··· a n := T a ··· a n ( X ) , the sets X a ··· a n form a sequence of nested ellipses. For each x ∈ E there exists aunique sequence a ∈ Σ := { , · · · , k } N such that π ( a ) := lim n →∞ T a ··· a n (0) = x where 0 denotes the origin. Let µ be a Bernoulli measure on Σ with associatedprobabilities p · · · p k . By a slight abuse of notation we also denote by µ themeasure µ ◦ π − on E . We wish to describe the scenery flow for µ .The collection A i of positive matrices defines a second iterated function systemon projective space. Given A i , we let φ i denote the action of A − i on PR , thatis φ i : PR → PR is such that a straight line passing through the origin at angle θ is mapped to a straight line through the origin at angle φ i ( θ ) by A − i . Sincethe matrices A i are strictly positive, the maps φ i strictly contract the negativequadrant Q of PR .For any θ ∈ Q and for any sequence a ∈ Σ the limitlim n →∞ φ a ◦ φ a ◦ · · · ◦ φ a n ( θ )exists and is independent of θ . There is a unique measure µ F on PR satisfying µ F ( A ) = k (cid:88) i =1 p i µ F ( φ i ( A )) . The measure µ F is called the Furstenberg measure and has been studied for ex-ample in [3].In our example of the previous section, the Furstenberg measure is a dirac masson direction − π corresponding to vertical projection, and the projection of µ inthis direction gave rise to a measure whose properties are key to understanding E λ . In our more general case of self affine sets E without a ‘carpet’ structure, µ F will typically have positive dimension, and the properties of projections of µ in µ F -almost every direction will be crucial.We say that a straight line is aligned in direction θ if it makes angle θ with thepositive real axis. or θ ∈ PR let π θ : E → [ − ,
1] denote orthogonal projection from E onto thediameter of unit disc X at angle θ , followed by the linear map from this diameterto [ − , µ θ on [ − ,
1] by µ θ := µ ◦ π − θ . Projection Condition:
We say that µ satisfies our projection condition if for µ F almost every θ ∈ PR the projected measure µ θ is absolutely continuous.In [8] it was shown that the Hausdorff, box and affinity dimensions of a self-affineset coincide if the natural Gibbs measure on E satisfies this projection condition.Furthermore, we gave a class of self-affine sets corresponding to an open set inparameter space for which the projection condition is satisfied, these exampleswere born out of the observation that the projection condition holds wheneverdim H µ F + dim H µ >
2, a condition which can often be shown to hold using roughlower bounds for dim H µ and dim H µ F .B´ar´any, Pollicott and Simon also gave regions of parameter space such that, foralmost every set of parameters in this region, the corresponding Furstenberg mea-sure is absolutely continuous [3]. Assuming absolute continuity of the Furstenbergmeasure, our projection condition holds whenever dim H µ > As a warm up to the later results describing the scenery flow for self-affine mea-sures, we begin by considering the scenery flow on slices through self-affine mea-sures in directions θ in the support of µ F . The results of this section do not requireany projection condition.Given θ ∈ PR , x ∈ [ − , µ θ,x of measures defined on theslices E θ,x := E ∩ π − θ ( x ) such that for each Borel set A ⊂ R we have µ ( A ) = (cid:90) [ − , µ θ,x ( A ∩ E θ,x ) dµ θ ( x ) . The family of slice measures µ θ,x is called the disintegration of µ . While the aboveequation does not uniquely define the family of measures µ θ,x , any two disintegra-tions of µ differ on a set of x of µ θ -measure 0. See [21] for more information ondisintegration of measures.Slicing measures can also be viewed as the limits of measures supported on thinstrips around the slice. Let E θ,x,(cid:15) denote the strip of width (cid:15) around the line E θ,x .Then for µ F almost every θ and µ θ almost every x , for any word a · · · a n we have µ θ,x ( X a ··· a n ∩ E θ,x ) = lim (cid:15) → µ ( X a ··· a n ∩ E θ,x,(cid:15) ) µ ( E θ,x,(cid:15) ) . (2)Let Σ ± := { , · · · , k } Z . Given a ∈ Σ ± we define the angle ρ ( a ) := lim n →∞ φ a ◦ φ a − ◦ · · · ◦ φ a − n ( θ ) or any θ ∈ Q . Then let π : Σ ± → Σ × Q be given by π ( a ) := ( a a a · · · , ρ ( a )) . We define a map f : Σ × PR → Σ × PR by f ( a, θ ) = ( σ ( a ) , φ a ( θ ))where σ is the left shift. Proposition 4.1.
The map f preserves measure µ × µ F . Furthermore, the system (Σ × PR , f, µ × µ F ) is ergodic. This was proved in [8]. The proof follows by observing that π is a continuous mapwhich factors (Σ ± , σ, µ ) onto (Σ × PR , f, µ × µ F ) and hence the ergodicity of σ passes to the factor map f .Our interest in the map f stems from its relevance to scaling scenery. The followingproposition is straightforward, and is proved in [8]. Proposition 4.2.
Let L ( a, θ ) denote the line passing through the element of E coded by a at angle θ . Then the map T − a : R → R maps the line L ( a, θ ) to theline L ( f ( a, θ )) . Our main result of this section is the following.
Theorem 4.1.
Let µ be a Bernoulli measure on a self-affine set E ⊂ R associatedto strictly positive matrices A i and satisfying our separation condition. Then thereexists a constant d such that for µ F -almost every θ ∈ PR and µ θ -almost every x ∈ [ − , the slice measure µ θ,x is exact dimensional with dimension d . The corresponding result for slices through non-overlapping self-similar measureswas proved by Hochman and Shmerkin [14], and this was extended to the over-lapping case by Falconer and Jin [10].We stress again that no condition on projections of the measure µ is required inthis section. The above result is an immediate corollary of the following theorem,which describes the scenery flow for the slice measures µ θ,x centered at points a ∈ E with π θ ( a ) = x . The constant d is the metric entropy of this flow.Let L ( a, θ, t ) denote the line at angle θ , centred at a and of length e − t . Let µ θ,a,t denote the measure µ θ,π θ ( a ) restricted to the line L ( a, θ, t ), linearly rescaled onto[ − ,
1] and renormalised to have mass 1.
Theorem 4.2.
There exists an ergodic fractal distribution P on the space of Borelprobability measures on [ − , such that for µ F almost every θ ∈ PR , for µ almostevery a ∈ Σ we have lim T →∞ T (cid:90) T δ µ θ,a,t dt → P. Theorem 4.2 implies Theorem 4.1 by a result of Hochman, see Proposition 1.19 of[14]. We prove Theorem 4.2. roof. First we need to verify that the self-affinity realtion for the measures µ carries over to a corresponding relationship between the measures µ θ,a,t . Given apoint ( a, θ ) we let r ( a, θ ) := inf { t : L ( a, θ, t ) ⊂ X } . Further, we let r ( a, θ ) := inf { t : L ( a, θ, t ) ⊂ X a } . Using equation 2 and noting that T − a ( X a ··· a n ∩ E θ,π θ ( a ) ,(cid:15) ) = X a ··· a n ∩ E φ a ( θ ) ,π φa θ ) ( σ ( a )) ,δ ( (cid:15) ) for some δ ( (cid:15) ) > (cid:15) →
0, we see that µ θ,a,r ( a,θ ) = µ φ a ( θ ) ,σ ( a ) ,r ( σ ( a ) ,φ a ( θ )) . The above equation says that, just as pieces of the slice through a at angle θ aremapped onto pieces of the slice through σ ( a ) at angle φ a ( θ ) by the map T − a ,so we can map pieces of the sliced measure onto their corresponding preimage.In particular, it allows us to understand the dynamics of zooming in on the slicemeasure µ θ,π a around a by relating small slices around a to larger slices around σ ( a ). We build a suspension flow that encapsulates these dynamics.Let roof function r : Σ × PR be given by r ( a, θ ) = r ( a, θ ) − r ( a, θ ). This isthe time taken to flow under φ from the line passing through a at angle θ andjust touching the boundary of X to the line centred at a, angle θ , touching theboundary of X a .Finally we let the flow ψ be the suspension flow over the system (Σ × PR , f ) withroof function given by r . That is, we define the space Z r := { (( a, θ ) , t ) : a ∈ Σ , θ ∈ PR , ≤ t ≤ r ( a, θ ) } where the points (( a, θ ) , r ( a, θ )) and ( f ( a, θ ) ,
0) are identified, and let the flow ψ s : Z r → Z r be given by ψ s (( a, θ ) , t ) := (( a, θ ) , s + t )for s + t ≤ r ( a, θ ), extending this to a flow for all positive time s by using theidentification (( a, θ ) , r ( a, θ )) = ( f ( a, θ ) , . We have already noted that the measure µ × µ F is f -invariant and ergodic. Thisgives rise to a ψ s -invariant, ergodic measure ν on Z r given by ν = ( µ × µ F × L ) | Z r where L denotes Lebesgue measure.There is an obvious factor map F from Z r to the space of Borel probability mea-sures on [ − ,
1] given by letting (( a, θ ) , t ) = µ a,θ,t . We have F ( ψ s ( a, θ, t )) = µ a,θ,t + s and thus we see that for µ × µ F almost every pair ( a, θ ) we have that the sceneryflow on the measure µ θ,a, generates the ergodic fractal distribution P = ν ◦ F − .This completes the proof of Theorem 4.2 and hence of Theorem 4.1.In essence, one can combine the work of sections 2 and 4 to give all the intuitionneeded to describe the scenery flow for the self-affine measures we consider. Whatfollows, which is occasionaly quite technical, verifies that this intuition is correct. Before describing the scenery flow, we describe a map from the space of measureson large ellipses to probability measures on X . This map plays the role of themap D of Observation 2, and will allow us to approximate the scenery flow on µ arbitrarily well.Given a ∈ Σ , θ ∈ PR , r , r > Y a,θ,r ,r be the ellipse centredat π ( a ), with long axis of length 2 e − r aligned in direction θ and with short axisof length 2 e − r .Given ( a, θ, r , r ) such that Y a,θ,r ,r (cid:54)⊂ X a , we let D a,θ,r ,r : Y a,θ,r ,r → X bethe bijection which maps the major axis of Y a,θ,r ,r to { } × [ − ,
1] and the minoraxis of Y a,θ,r ,r to [ − , × { } .Let D a,θ,r ,r also denote the analagous map which maps finite measures on ellipses Y a,θ,r ,r to probability measures on X . As in observation 3 of Section 2, weconsider what happens to the family of dilated measures as the minor axis of anellipse shrinks. Lemma 5.1.
Let a ∈ Σ , θ ∈ PR , r > . Suppose that for µ a,θ,r almost every b ∈ Σ we have that there exist infinitely many n ∈ N such that the projection of µ | X b ··· bn in direction θ is absolutely continuous, and that the scenery flow on thisprojected measure centred at π θ ( b ) converges to Lebesgue measure. Then we have lim r →∞ D a,θ,r ,r ( µ | Y ( a,θ,r ,r ) ) = ( L × µ a,θ,r ) | X ( L × µ a,θ,r )( X ) . Proof.
Our notion of convergence here is that of the Prokhorov metric. It isenough to show that, for all N ∈ N , we can divide the unit square into a grid of2 N + 1 squares A i,j of equal side length and have that for each i, j ∈ {− N, · · · , N } such that A i,j ⊂ X , (cid:0) D a,θ,r ,r ( µ | Y ( a,θ,r ,r ) ) (cid:1) ( A i,j ) → ( L × µ a,θ,r ) | X ( L × µ a,θ,r )( X ) ( A i,j ) irst we consider squares A ,j whose x -coordinate is at the origin. Since slicingmeasures are almost surely the limit of the measures µ restricted to a thin striparound the slice, we have that the relative distribution of mass within the squares A ,j converges to the slicing measure µ a,θ,r as r → ∞ (given θ this holds for µ -almost every a .Now we fix j and consider the horizontal distribution of mass in µ | Y ( a,θ,r ,r ) (cid:16) D − a,θ,r ,r ( A i,j ) (cid:17) µ ( Y ( a, θ, r , r ))for i varying.We note that Y ( a, θ, r , r ) intersects various ellipses. The ellipses X b ··· b m areseparated, and as r → ∞ the angle of the strip D a,θ,r ,r ( X b ··· b m ∩ Y ( a, θ, r , r ))tends to the horizontal (indeed, any line which is not in direction θ gets pulledtowards the horizontal by D a,θ,r ,r , and as r → ∞ this effect becomes ever morepronounced). For m large enough, each strip D a,θ,r ,r ( X b ··· b m ∩ Y ( a, θ, r , r )) iscontained within the horizontal rectangle ∪ Ni = − N A i,j for some j ∈ {− N, · · · , N } . T he m ap D Figure 2: The action of D on an ellipse Y . We want to understand the distribution of mass horizontally within the rectangles ∪ Ni = − N A i,j .We will consider the projection of µ | X b ··· bm in direction θ , intersected with Y ( a, θ, r , r ),dilated to be supported on [ − ,
1] and normalised to have mass 1. This is just thescenery flow on the projection of µ | X b ··· bm in direction θ , centred at π θ ( b ) at time r . ow we assumed that π θ ( µ | X b ··· bm ) was absolutely continuous with positive den-sity at π θ ( b ). Then by lemma 2.1 this scenery flow converges to Lebesgue measure.Thus the horizontal distribution of mass within the rectangles ∪ mi = − m ( j ) ( j ) A i,j con-verge to Lebesgue measure as r → ∞ for all j , where m ( j ) is the largest naturalnumber such that A m ( j ) ,j ⊂ X . Then we are done.This yields the following corollary. Corollary 5.1.
Suppose that our projection condition holds, i.e. that the projectedmeasure µ θ is absolutely continuous for µ F almost every θ ∈ PR . Then for µ -almost every a , µ F almost every θ and all r we have lim r →∞ D a,θ,r ,r ( µ | Y ( a,θ,r ,r ) ) = ( L × µ a,θ,r ) | X ( L × µ a,θ,r )( X ) . Proof.
First we note, using our affinity relation, that the measure obtained byprojecting µ | X b ··· bn in direction θ ∈ supp ( µ F ) centred at π θ ( b ) is a scaled downcopy of the measure obtained by projecting µ in direction φ b n ◦ · · · ◦ φ b ( θ ) centredat π φ bn ◦···◦ φ b ( θ ) ( σ n ( b )), see [8] for a careful proof.Since the directions θ are distibuted according to µ F , for µ -almost every b itfollows that the projected measure µ φ bn ◦···◦ φ b ( θ ) is absolutely continuous for all n and that the scenery flow centred at π φ bn ◦···◦ φ b ( θ ) ( σ n ( b )) converges to Lebesguemeasure.If a condition holds for µ × µ F -almost every ( b, θ ) then it follows that for all r > µ × µ F almost every ( a, θ ) the condition holds for µ a,θ,r almost every b . Thenwe see that the hypotheses of Lemma 5.1 hold, and so the conclusions hold also,as required. Given a pair ( a, t ) we let n = n ( a, t ) = max { n ∈ N : B ( a, e − t ) ⊂ X a ··· a n } . Then we associate to small ball B ( a, e − t ), coupled with angle θ , the ellipse Y σ n ( a ) ,φ an ◦···◦ φ a ( θ ) ,t +log( α ( a ··· a n )) ,t +log( α ( a ··· a n )) . Note that the first two parameters here are equal to f n ( a, θ ). Since α ( a · · · a n ) → t, n → ∞ , log( α ( a · · · a n )) is negative. In fact, the quantity t +log( α ( a · · · a n ))remains bounded as t → ∞ .Let ν ( a, θ, t ) := D f n ( a,θ ) ,t +log( α ( a ··· a n )) ,t +log( α ( a ··· a n )) ( µ | Y fn ( a,θ ) .t +log( α a ··· an )) ,t +log( α a ··· an )) ) , he measures ν ( a, θ, t ) are elements of M .We define a map F which takes probability measures on [ − ,
1] to probabilitymeasures on X by F ( m ) := ( L × m ) | X ( L × m )( X ) . Let the distribution P on the space of Borel probability measures on X be theimage of P under F , where P was defined in Section 4. Then the two dimensionalscenery flow on ( M , P ) is a factor of the one dimensional scenery flow on ( M , P )under the factor map F , and so it follows immediately that P is an ergodic fractaldistribution, since ergodicity passes to factors of ergodic systems. One can readilyverify that the distribution P is quasi-Palm. Theorem 5.1.
For µ × µ F almost every pair ( a, θ )lim T →∞ T (cid:90) T ν ( a, θ, t ) dt = P . Proof.
Let ( a, θ ) be such that the sliced scenery flow on the measure µ a,θ, gen-erates P , and such that f n ( a, θ ) satisfies the conditions of Corollary 5.1 for each n ∈ N . The set of ( a, θ ) for which this holds has µ × µ F measure one, since it is acountable intersection of sets of measure 1.For (cid:15), N >
0, let the bad set B ( (cid:15), N ) be given by B ( (cid:15), N ) := (cid:26) ( a, θ ) : (cid:12)(cid:12)(cid:12)(cid:12) D a,θ,r ,r ( µ | Y ( a,θ, ,r ) ) − ( L × µ a,θ, ) | X ( L × µ a,θ, )( X ) (cid:12)(cid:12)(cid:12)(cid:12) > (cid:15) for some r > N (cid:27) . Then for all (cid:15), (cid:15) >
0, using Egorov’s theorem and Lemma 5.1, there exist
N > µ × µ F )( B ( (cid:15), δ )) < (cid:15) . Now note that for any a there exists a T such that T − a ··· a n ( B ( π ( a ) , e − t )) is anellipse with minor axis of length less than e − N for all t > T . Then since thescenery flow on ( a, θ,
1) generates P , and since (cid:15) , N were arbitrary, we see thatlim T →∞ T (cid:90) T ν ( a, θ, t ) dt = P . as required.Finally we state a continuity result. The proof of this result requires a little geom-etry, and is most likely of limited interest, and so can be found in the appendix. Proposition 5.1.
For each a ∈ Σ , t ∈ R the map θ → ν a,θ,t is continuous in θ and this continuity is uniform over t . in particular, for all θ ∈ PR and for all (cid:15) > there exists δ > such that if | θ − θ (cid:48) | < δ then for allsubsets A ⊂ M we have (cid:12)(cid:12)(cid:12)(cid:12) lim T →∞ T L{ t ∈ [0 , T ] : ν ( a, θ, t ) ∈ A } − lim T →∞ T L{ t ∈ [0 , T ] : ν ( a, θ (cid:48) , t ) ∈ A } (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15), and so the sceneries generated by ν ( a, θ, t ) and ν ( a, θ (cid:48) , t ) are close. The Full Scenery Flow
We now relate the scenery flow on µ to the measures ν of the previous section.We let S t,a denote the bijective linear map from B ( π ( a ) , e − t ) to X given by ex-panding all vectors by e t and translating the resulting ball to the origin. We alsolet S t,a be the scenery flow map from finite measures on B ( π ( a ) , e − t ) to probabil-ity measures on X . In this section we first describe the preimages of small ballsunder maps T − a ··· a n , and then decompose the scenery flow for µ using the maps D of the previous section.The fact that we are considering only strictly positive matrices leads to somesimple observations about the intersection of B ( x, r ) with the self-affine set E .Let α ( a · · · a n ), α ( a · · · a n ) denote the lengths of the major and minor axes ofthe ellipse X a ··· a n . Then the ratio α ( a ··· a n ) α ( a ··· a n ) tends to 0 as n → ∞ at some uniformrate independent of a (see [8]).There exists a H¨older continuous function F : Σ → PR such that, for each a ∈ Σ,the ellipses X a ··· a n are aligned so that the angle that their long axis makes withthe x -axis tends to F ( a ) as n → ∞ . This convergence is uniform over a ∈ Σ. Infact, F ( a ) is given by F ( a ) := lim n →∞ φ − a ◦ · · · ◦ φ − a n (0) ∈ Q . The strong stable foliation, which gives the limiting direction of the minor axis ofellipses X a ··· a n , is given by F ss ( a ) := lim n →∞ φ a ◦ · · · ◦ φ a n (0) ∈ Q . Note that F ( a ) and F ss ( a ) are perpendicular.Let θ ( a · · · a n ) ∈ Q be the direction of the minor axis of the ellipse X a ··· a n . Proposition 6.1.
Let e − t < α ( a · · · a n ) . Then T − a ··· a n ( B ( π ( a ) , e − t )) is an ellipse centred at π ( σ n ( a )) with major axis of length e − t . ( α ( a · · · a n )) − aligned in direction φ a n ◦ · · · φ a ( θ ( a · · · a n )) and minor axis of length equal to e − t . ( α ( a · · · a n )) − . Stated using our notation for ellipses, this says T − a ··· a n ( B ( π ( a ) , e − t )) = Y σ n ( a ) ,φ an ◦··· φ a ( θ ( a ··· a n )) ,t +log( α ( a ··· a n )) ,t +log( α ( a ··· a n )) . Proof.
Lines which bisect the ellipse X a ··· a n just touching the edges and passingthrough the centre are mapped by T − a ··· a n to lines passing through the origin whichjust touch the boundary of the unit disk. This fact allows us to see how much thelinear map T − a ··· a n expands different lines. n particular, the maximal expansion rate is on lines in direction θ ( a · · · a n ),parallel to the minor axis of X a ··· a n . These are expanded linearly by a factor α ( a ··· a n ) , and by the definition of φ i we see they are mapped to direction φ a n ◦· · · φ a ( θ ( a · · · a n )), note the reversed order of the word a · · · a n here.The major axis of X a ··· a n gives rise to the smallest expansion rate of the map T − a ··· a n , which is α ( a ··· a n ) , thus the minor axis of T − a ··· a n B ( π ( a ) , e − t )) has length e − t . ( α ( a · · · a n )) − .We now discuss functions which map our ellipses T − a ··· a n ( B ( π ( a ) , r )) to the unitdisk. Note that any bijective linear map from B ( π ( a ) , r ) to X which maps π ( a )to the origin and which preserves the directions F ( a ) and F ss ( a ) must be thesame as our dilation map S − log r,a . This is because a linear map in R is uniquelydetermined by its action on any two vectors which span R . Proposition 6.2.
We have S t,a = R θ ⊥ ( a ··· a n ) ◦ D σ n ( a ) ,φ an ◦··· φ a ( θ ( a ··· a n )) ,t +log( α ( a ··· a n )) ,t +log( α ( a ··· a n ) ◦ T − a ··· a n where n = n ( a, t ) is such that n is the largest natural number for which B ( π ( a ) , e − t ) ⊂ X a ··· a n .Proof. The previous proposition noted that T − a ··· a n ( B ( π ( a ) , e − t )) is an ellipse cen-tred at π ( σ n ( a )). It also follows from the proof that T − a ··· a n maps lines in di-rection θ ( a · · · a n ) to lines in direction φ a n ◦ · · · φ a ( θ ( a · · · a n )). Furthermore,the perpendicular angles of the major and minor axis of the ellipse X a ··· a n aremapped on to the perpendicular angles of the minor and major axis of the ellipse T − a ··· a n ( B ( π ( a, e − t ))).Then D σ n ( a ) ,φ an ◦··· φ a ( θ ( a ··· a n )) ,t +log( α ( a ··· a n )) ,t +log( α ( a ··· a n ) ◦ T − a ··· a n maps B ( a, e − t ) bijectively onto X , where the diameter of B ( a, e − t ) at angle( θ ( a · · · a n )) is mapped to { } × [ − ,
1] and the diameter at angle θ ⊥ ( a · · · a n ) ismapped to [ − , × { } .Rotating by angle θ ⊥ ( a · · · a n ) we see that the image of the major and minor axesof X a ··· a n are oriented in the correct direction.Then we see that our map is a bijective map from B ( π ( a ) , e − t ) to X which main-tains the directions θ ( a · · · a n ) and θ ⊥ ( a · · · a n ), so we are done.In particular, this yields the following theorem. Theorem 6.1.
Let µ be a Bernoulli measure on a self-affine set E associated tostrictly positive matrices, and assume that for µ F almost every θ ∈ PR the image µ θ of µ under projection in direction θ is absolutely continuous. Then for µ -almostevery a the scenery flow S t,a ( µ ) is given by S t,a ( µ ) = R θ ⊥ ( a ··· a n ) ( ν a,θ ( a ··· a n ) ,t ) . s t → ∞ this flow is asymptotic to the flow R F ( a ) ( ν a,F ss ( a ) ,t ) and so generates the ergodic fractal distribution R F ( a ) ◦ P . By R F ( a ) ◦ P we mean the distribution on M obtained by picking measures µ ∈ M according to P and then rotating the resulting measure by angle R F ( a ) .Hence we see that µ is not a uniformly scaling measure, unless the foliation F ( a )gives the same angle for each a . This happends only when the maps φ i all havea common fixed point, in which case the Furstenberg measure µ F is a Dirac massand the corresponding self-affine set has a carpet like construction. In particular,Theorem 2.1 is a corollary to this theorem.Finally we comment that one does not automatically have that for µ almost every a the flow ν a,F ss ( a ) ,t equidistributes with respect to P , since there is an obviousdependence between a and F ss ( a ). Here we rely on our continuity proposition(Proposition 5.1) which allows us to replace F ss ( a ) with µ F -typical angles θ closeto F ss ( a ) such that the distance between the orbits ν a,θ,t and ν a,F ss ( a ) ,t remainssmall. Proof.
First we note that, by the ergodic theorem, for µ almost every a and forall (cid:15) > θ ∈ ( F ss ( a ) − δ, F ss ( a ) + δ ) such that the family of measures ν a,θ,t equidistributes with respect to P . Now since θ ( a · · · a n ) → F ss ( a ) we seethat the sequence θ ( a · · · a n ) is eventually bounded within distance 2 δ of θ . Thenby Proposition 5.1 we have that the measures ν a,θ ( a ··· a n ) ,t and ν a,θ,t are within (cid:15) of each other, and so, since (cid:15) was arbitrary, we have that the family of measures ν a,θ ( a ··· a n ) ,t generate P .Finally, incorporating the rotation element and using Proposition 6.2 we have that S t,a ( µ ) generates the distribution R F ( a ) ◦ P . Despite having been worked on for over 25 years, a general theory of the dimensionof self-affine sets has proved ellusive. Indeed, questions such as whether boxdimension always exists for self-affine sets remain open. The scenery flow seemslike a natural tool to transfer results from ergodic theory to the study of dimensionfor self-affine sets.There are a number of further questions which could lead towards a more generaltheory of scenery flow for self-affine sets.
Question 1:
Can one conclude that examples of section 2 uniformly scaling mea-sures generating ergodic fractal distributions whenever the corresponding Bernoulliconvolution is a uniformly scaling measure generating an ergodic fractal distribu-tion? uestion 2: Are overlapping self-similar sets uniformly scaling measures generat-ing ergodic fractal distributions? What about projections of self-affine sets? Thesecond part will most likely follow from the first, given the dynamical structureof projections of self-affine sets described in [8].
Question 3:
Suppose that for µ F almost every θ the projection π θ : E → [ − , µ is a uniformlyscaling measure generating an ergodic fractal distribution?Finally we comment on the condition that the matrices generating our self-affineset should be strictly positive. This condition ensures that the maps φ i strictlycontract the negative quadrant and hence that µ F can be defined via an iteratedfunction system construction. The condition is also useful in making a lot ofconvergence results uniform. It seems likely that the condition can be relaxed.The Furstenberg measure can be defined without any cone condition, see [6]. θ Lemma 8.1. α ( a · · · a n ) α ( a · · · a n ) tan( φ a ··· a n ( θ ) − φ a ··· a n ( θ ( a · · · a n ))) = tan( θ − θ ( a · · · a n )) Proof.
The linear map A − a ··· a n stretches lines at angle θ ( a · · · a n ) by α ( a · · · a n ) − and lines at angle θ ( a · · · a n ) ⊥ by α ( a · · · a n ) − . The lemma follows using basicgeometry.We now consider when one ellipse can fit inside an expanded, rotated concentriccopy of itself. Lemma 8.2.
Let Y be an ellipse centred at the origin with major and minoraxes of length α , α respectively and with major axis oriented along the y -axis.Let Z be an ellipse centred at the origin with major and minor axes of length (1 − (cid:15) ) α , (1 − (cid:15) ) α respectively with major axis oriented at angle θ from the vertical.Then Z ⊂ Y whenever α α tan( θ ) < − (cid:15) − .Proof. The line from the origin to the boundary of Y at angle ρ has length α cos( ρ ) + α sin( ρ ). The corresponding line for ellipse Z has length(1 − (cid:15) )( α (cos( ρ − θ )) + α (sin( ρ − θ )))= (1 − (cid:15) )( α (cos( ρ ) cos( θ ) + sin( ρ ) sin( θ )) + α (sin( ρ ) cos( θ ) − cos( ρ ) sin( θ ))) ≤ (1 − (cid:15) )( α cos( ρ ) + α sin( ρ ))(cos( θ ) + α α (sin( θ )) . So if we have cos( θ ) + α α sin( θ ) ≤ − (cid:15) hen we will have for each angle ρ that the slice through Z at angle ρ is a subsetof the slice through Y at angle ρ , and hence that Z ⊂ Y . The above inequalityholds whenever α α tan( θ ) < − (cid:15) − Lemma 8.3. let a ∈ Σ and suppose that θ is such that | tan( θ ) − tan( F ss ( a )) | < − (cid:15) − . Then Y f n ( a,θ ) , − log( α ( a ··· a n ) − (cid:15), − log( α ( a ··· a n )) − (cid:15) ⊂ Y f n ( a,θ ( a ··· a n )) , − log( α ( a ··· a n )) , − log( α ( a ··· a n ))) for all large enough n .Proof. By lemma 8.1 we have that α ( a · · · a n ) α ( a · · · a n ) tan( φ a ··· a n ( θ ) − φ a ··· a n ( θ ( a · · · a n ))) = tan( θ − θ ( a · · · a n )) < − (cid:15) − θ ( a · · · a n ) → F ss ( a ). Then by lemma 8.2 we are done.We now consider our maps D which dilate ellipses. We show that if Z ⊂ Y withthe area of Y close to that of Z then the measure D Z ( µ | Z ) is close to D Y ( µ | Y ).We do this by showing that the natural magnification map D Z from Z to the unitdisk is the same as first magnifying Z using the magnification map D Y on Y toget some other ellipse W ⊂ X , and then using the magnification map D W on W . Lemma 8.4.
Let Y a,θ,r ,r ⊂ Y a,θ (cid:48) ,r (cid:48) ,r (cid:48) . Let a (cid:48)(cid:48) , θ (cid:48)(cid:48) , r (cid:48)(cid:48) , r (cid:48)(cid:48) be such that D a,θ (cid:48) ,r (cid:48) ,r (cid:48) ( Y a,θ,r ,r ) = Y a (cid:48)(cid:48) ,θ (cid:48)(cid:48) ,r (cid:48)(cid:48) ,r (cid:48)(cid:48) ⊂ X. Then D a,θ,r ,r = D a (cid:48)(cid:48) ,θ (cid:48)(cid:48) ,r (cid:48)(cid:48) ,r (cid:48)(cid:48) ◦ D a,θ (cid:48) ,r (cid:48) ,r (cid:48) : Y a,θ,r ,r → X. Proof.
Since the map D a (cid:48)(cid:48) ,θ (cid:48)(cid:48) ,r (cid:48)(cid:48) ,r (cid:48)(cid:48) ◦ D a,θ (cid:48) ,r (cid:48) ,r (cid:48) : Y a,θ,r ,r → X is bijective andmaps the major and minor axes onto the vertical and horizontal axes, this isimmediate. Lemma 8.5.
For all (cid:15) > there exists δ > such that whenever ellipse W ⊂ X has area larger than − δ and long axis oriented within δ of the vertical then themap D W : W → Z is within (cid:15) of the identity map. This is again immediate.Putting all of the previous lemmas together yields the following theorem.
Theorem 8.1.
For all (cid:15) > there exists δ > such that, for all θ with | F ss ( a ) − θ | < δ we have that d (cid:16) ν a,θ,t , D σ n ( a ) ,φ an ◦··· φ a ( θ ( a ··· a n )) ,t +log( α ( a ··· a n )) ,t +log( α ( a ··· a n )) ◦ T − a ··· a n ( µ | B ( π ( a ) ,e − t ) ) (cid:17) < (cid:15). This continuity theorem allows one to use the flow giving rise to measures ν toinfer properties of the scenery flow. cknowledgements Many thanks to Jon Fraser and Kenneth Falconer for many useful discussions.This work was supported by the EPSRC, grant number EP/K029061/1.
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