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Bernoulli Convolutions and 1D Dynamics Tom Kempton & Tomas Persson 5 1 January 28, 2015 0 2 n a Abstract J 7 2 We describe a family φ of dynamical systems on the unit interval λ ] which preserve Bernoulli convolutions. We show that if there are pa- S rameter ranges for which these systems are piecewise convex, then D the corresponding Bernoulli convolution will be absolutely continuous . h with bounded density. We study the systems φ and give some nu- t λ a merical evidence to suggest values of λ for which φ may be piecewise m λ convex. [ 1 v 0 1 Introduction 4 7 6 0 In the study of self similar measures corresponding to non-overlapping iter- . 1 ated function systems, there is a natural way of defining an expanding dy- 0 5 namical system which preserves the measure and which allows one to study 1 : various properties such as dimension. The case of self similar measures with v i overlaps is much more involved, and it is not clear how best to study them X using dynamical systems. r a Bernoulli convolutions are a particularly well studied family of self-similar measures. For each λ ∈ (0,1) we define the corresponding Bernoulli convolution ν to be the distribution of the series λ ∞ (cid:88) (λ−1 −1) a λi i i=1 1 where the digits a are picked independently from digit set {0,1} with prob- i ability 1. Equivalently, Bernoulli convolutions are the unique probability 2 measures satisfying the self similarity relation 1 ν = (ν ◦T +ν ◦T ), λ λ 0 λ 1 2 where the maps T : R → R are defined by T (x) = x − (λ−1 − 1)i. For i i λ λ ∈ (0, 1), the self similar measures are generated by a non-overlapping 2 iterated function system, and are invariant under the interval maps φ given λ by  λ−1x x ∈ [0,λ]  φ (x) = 1 x ∈ (λ,1−λ) . λ  1−λ−1(x−(1−λ)) x ∈ (1−λ,1) 0 1 0 1 0 1 Figure 1: The maps φ for λ equal to 0.2, 0.4 and 0.5 respectively λ Themainaimofthisarticleistoextendthedefinitionofφ totheoverlapping λ case, when λ ∈ (1,1), and to study ν using these interval maps. 2 λ There are a number of long standing open questions about Bernoulli convolutions, chief among which is the question of for which parameters λ the corresponding measure ν is absolutely continuous. It is known that each λ Bernoulli convolution is either purely singular or absolutely continuous, see [8]. IfλistheinverseofaPisotnumberthenν issingular, see[5], andinfact λ has Hausdorff dimension less than one, [9]. In [7] Garsia gave a small, explicitly defined class of algebraic integers for which ν is known to be absolutely λ continuous. Solomyak proved in [13] that ν is absolutely continuous for al- λ most every λ ∈ (1,1), and in [12] Shmerkin proved that the set of λ ∈ (1,1) 2 2 admittingsingular Bernoulli convolutions hasHausdorffdimension 0, but the question of determining the parameters λ which admit absolutely continuous Bernoulli convolutions remains open. 2 There are other interesting open questions regarding Bernoulli convolutions. For example, is it the case that any singular Bernoulli convolution must have Hausdorff dimension less than one? Do there exist intervals in the parameter space for which every Bernoulli convolution is absolutely continuous (and even has continuous density)? Does the density evolve continuously with λ? Similar questions exist in the study of invariant measures associated to various one parameter families of interval maps, and in this area a good deal of progress has been made [4, 6, 10]. With this in mind, we extend the definition of the generalised tent maps φ to the overlapping case. These λ tent maps preserve the corresponding Bernoulli convolutions ν . They are λ described implicitly in terms of the distribution F of ν , and while we are λ λ able to write down explicit formulae for the φ only in some special cases, λ we are able to prove some general properties. In particular, we prove that if φ is piecewise convex for all λ in some interval λ (a,b) then the corresponding Bernoulli convolution is absolutely continuous with bounded density. For each x ∈ [0,1] the map x (cid:55)→ φ (x) is continuous λ in λ, and convexity is preserved by passing to limits in a continuous family of functions. Thus, piecewise convexity of the functions φ seems like an λ appropriate vehicle for passing from almost everywhere absolute continuity to everywhere absolute continuity for parameters in certain ranges. We can show that φ is piecewise convex for certain special cases, and remain opti- λ misticthatonemaybeabletoproveanalyticallythatthemapφ ispiecewise λ convex in certain parameter ranges. For the moment however, our results on piecewise convexity are restricted to some special values of λ, although we are able to run numerical approximations for any λ. There have been previous numerical investigations into Bernoulli convolutions, we mention in particular the work of Benjamini and Solomyak [1] and of Calkin et al [2, 3]. In the next section we define the maps φ in which we are interested and λ prove that they preserve Bernoulli convolutions. We prove some elementary properties of the maps φ and give the maps explicitly in some special cases. λ In section 3 we prove various properties of ν that would follow from φ λ λ being piecewise convex, and in section 4 we give some numerical evidence on the piecewise convexity of φ . Finally in section 5 we state some further λ questions and conjectures. 3 2 Generalised Tent Maps Let F : [0,1] → [0,1] be the distribution of ν , i.e. F (x) := ν [0,x]. F λ λ λ λ λ is strictly increasing because ν is fully supported. We define a map φ : λ λ [0,1] → [0,1] by (cid:26) F−1(2F (x)) x ∈ [0, 1] φ (x) = λ λ 2 . λ F−1(2F (1−x)) x ∈ [1,1] λ λ 2 Since F is strictly increasing on [0,1], the map φ is well defined. We will λ λ see later that φ preserves ν . λ λ y y y x x x Figure 2: Graphs of φ for λ = 0.6,0.7 and 0.8. λ The map φ will be the chief object of study for this article. Since F can λ λ be well approximated numerically, to known levels of accuracy, one can gain good numerical approximations to the maps φ . Three such approximations λ are displayed for different values of λ in Figure 2. We begin by observing some simple properties of φ . λ Lemma 2.1. The map φ has the following properties for λ ∈ (1,1). λ 2 1. φ (0) = φ (1) = 0. λ λ (cid:0) (cid:1) 2. φ 1 = 1. λ 2 (cid:2) (cid:3) (cid:2) (cid:3) 3. φ is strictly increasing on 0, 1 and strictly decreasing on 1,1 . λ 2 2 4. φ (x) = φ (1−x). λ λ 5. φ is continuous. λ 4 Proof. We have that F (0) = 0,F (1) = 1 and F (1) = 1 because ν is λ λ 2 2 λ λ supported on [0,1] and symmetric about the point 1. Then points 1 and 2 2 follow immediately. Part 4 can be seen to be true by looking at the piecewise definition of φ . λ Becauseν [a,b] > 0foreach0 ≤ a < b ≤ 1wehavethatF isstrictlyincreas- λ λ (cid:2) (cid:3) ing. Consequently φ is strictly increasing on 0, 1 (and strictly decreasing λ 2 on [1,1]). 2 Finally, we observe that continuity of φ follows from the fact that ν is non- λ λ atomic and that ν [a,b] > 0 for each 0 ≤ a < b ≤ 1. Then both F and F−1 λ λ λ are uniformly continuous, and so φ is continuous in x. λ We call maps satisfying the above properties generalised tent maps. Our first theorem is the following Theorem 2.1. Let λ ∈ (1,1). Then ν is invariant under φ . 2 λ λ Proof. It is enough to show that for each a ∈ [0,1] we have that ν [a,1] = ν (φ−1[a,1]). λ λ λ To prove this, we note that φ−1[a,1] = [b,1−b] λ where b ∈ [0, 1] satisfies φ (b) = a. But then 2 λ 1 ν [b,1−b] = 2ν [b, ] λ λ 2 1 = 2(F ( )−F (b)) λ λ 2 = 1−2F (b) λ = F (1)−F (φ (b)) λ λ λ = F (1)−F (a) λ λ = ν [a,1], λ as required. Thus, if ν is absolutely continuous we see that φ has an acip. We have not λ λ been able to prove the converse statement, that φ does not have an acip in λ 5 the case that ν is singular, this would be a useful statement which would λ make the relationship between the study of φ and the measures ν a little λ λ more straightforward. The following theorem shows that the maps φ evolve continuously in λ. λ Theorem 2.2. For each x ∈ [0,1],λ ∈ (1,1) we have that φ (x) → φ (x) 0 2 λ λ0 as λ → λ . 0 Proof. Fix λ ∈ (1,1). We rely on three facts for this proof. 0 2 Firstly we use that the function F−1 is continuous in x: for all (cid:15) > 0 there λ 2 exists (cid:15) > 0 such that 1 |x−y| < 2(cid:15) =⇒ |F−1(x)−F−1(y)| < (cid:15) . (1) 1 λ λ 2 Secondly we use that for each x ∈ [0,1] the function F (x) is continuous in λ λ: for all (cid:15) > 0 there exists δ > 0 such that 1 1 |λ−λ | < δ =⇒ |F (x)−F (x)| < (cid:15) . (2) 0 1 λ λ0 1 Finally we use that for each x ∈ [0,1] the function F−1(x) is continuous in λ λ. For all (cid:15) > 0 there exists a δ > 0 such that 3 2 |λ−λ | < δ =⇒ |F−1(x)−F−1(x)| < (cid:15) . (3) 0 2 λ λ0 3 We fix x and let δ = min{δ ,δ } and |λ−λ | < δ. Then 1 2 0 |φ (x)−φ (x)| = |F−1(2F (x))−F−1(2F (x))| λ λ0 λ λ λ0 λ0 ≤ sup |F−1(2F (x))−F−1(y)| λ λ λ0 2Fλ(x)−2(cid:15)1≤y≤2Fλ(x)+2(cid:15)1 ≤ |F−1(2F (x))−F−1(2F (x))|+(cid:15) λ λ λ0 λ 2 ≤ (cid:15) +(cid:15) , 3 2 Here the second line holds since 2F (x) ∈ (2F (x) − 2(cid:15),2F (x) + 2(cid:15)) by λ0 λ λ equation 2. Then the third and fourth line follows from equations 1 and 3 respectively. Since (cid:15) ,(cid:15) were arbitrary, we are done. 2 3 In the case that one knows the distribution F , one can write down the map λ φλ explicitly. In particular, for λ = 2−n1, it is not difficult to write down φλ. 6 y y y =F (x) λ y =φ (x) λ x x √ Figure 3: Graphs of F and φ for λ = 1/ 2. λ λ Example 2.1. In the case λ = √1 , F is given by 2 λ  √ (cid:104) (cid:105)  (3 2+1)x2 x ∈ 0, 1√  4 √ 1+ 2√ Fλ(x) = (1+ √1 )x− 2 x ∈ [ 1√ , √2 ] .  2 √ 4 1+√ 2 1+ 2  1−(1+ 3 2)(1−x)2 x ∈ [ √2 ,1] 4 1+ 2 Consequently φ is given by λ  √  2x x ∈ [0, 1√ ]  √ (cid:104) 2+ 2 (cid:105)  φ (x) = (1+ 2)x2 + 1√ x ∈ 1√ , 1√ λ 2+2 2 2+ 2 1+ 2  1−2(cid:16)1/2√−x(cid:17)1/2 x ∈ (cid:104) 1√ , 1(cid:105) 1+ 2 1+ 2 2 which is extended to the whole interval I using the symmetry around 1. We λ 2 have drawn the graphs of F and φ in Figure 3. λ λ 2.1 Further properties of φ λ Whilewecannotwritedownφ explicitly, wecandescribethebehaviournear λ x = 0 and the rate of the blowup at x = 1. The following lemma describes 2 φ near 0, and hence also the behaviour near 1. λ Lemma 2.2. We have that φ (x) = λ−1x λ for x ∈ [0,1−λ]. 7 Proof. Self similarity of the measures ν give that λ 1 (cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1) F (x) = F λ−1x +F λ−1x−(λ−1 −1) (4) λ λ λ 2 Then F (φ (x)) = 2F (x) = F (λ−1x)+F (λ−1x−(λ−1 −1)). λ λ λ λ λ But because F (x) = 0 for x ≤ 0, we have λ F (λ−1x−(λ−1 −1)) = 0 λ for x ≤ 1−λ. Then F (φ (x)) = 2F (x) = F (λ−1x), λ λ λ λ for x ∈ [0,1−λ], which completes the proof. (cid:2) (cid:3) It remains to find φ (x) for x ∈ 1−λ, 1 , and then by symmetry to define λ 2 φ on [1,1]. We can also describe the nature of φ around x = 1 for typical λ 2 λ 2 λ. Lemma 2.3. We have that (cid:18) (cid:19) φ 1 −x ≈ 1−cx−llooggλ2 2 for small x, where c is a constant that depend continuously on λ. Proof. We start by noting that, since φ (x) evolves continuously in λ, it is λ enough to describe the nature of the blowup for values of λ corresponding to absolutely continuous ν , as by passing to limits we get the result for all λ. λ We consider the behaviour of F (x) close to x = 1 and to x = 1. Assuming λ 2 that h (1) exists and is positive, we have that λ 2 (cid:18) (cid:19) (cid:18) (cid:19) 1 1 1 F −(cid:15) ≈ F −h (cid:15) = −h (cid:15). λ λ λ λ 2 2 2 Thus we have that (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)1 1 1 (cid:12) 1 (cid:12)1 1 (cid:12) (cid:12) −F ( − (cid:15))(cid:12) ≈ (cid:12) −F ( −(cid:15))(cid:12). (5) λ λ (cid:12)2 2 2 (cid:12) 2 (cid:12)2 2 (cid:12) 8 Conversely, equation 4 gives that for small δ 1 F (1−δ) = (1+F (λ−1(1−δ)−(λ−1 −1))) λ λ 2 1 1 = + F (1−λ−1δ) λ 2 2 giving 1 1−F (1−δ) = (1−F (1−λ−1δ)). (6) λ λ 2 (cid:0) (cid:1) Now suppose that φ 1 −(cid:15) = 1 − δ for some fixed (cid:15) and δ. Then by λ 2 equations 5 and 6 we have that 1 1 φ ( − (cid:15)) ≈ 1−λδ. λ 2 2 Iterating, we have (cid:18) (cid:18) (cid:19)n (cid:19) 1 1 φ − (cid:15) ≈ 1−λnδ, λ 2 2 and we see that we have a blow up of the form (cid:18) (cid:19) φ 1 −x ≈ 1− δ x−llooggλ2 λ 2 ε−logλ/log2 with c = δεlogλ/log2 depending continuously on λ. 3 Piecewise Convexity Numerical approximations of the maps φ suggest that there are ranges of λ λ close to 1 in which the maps φ are piecewise convex. For the value λ = √1 , λ 2 one can see directly from the calculation of the previous section that φ is λ piecewise convex. In this section we prove various properties of ν that would follow from λ φ | being convex. We use the term ‘piecewise convex’ as shorthand for λ [0,1] 2 the statement that φ is convex on each of the two intervals [0, 1] and [1,1]. λ 2 2 The following theorem shows the relevance of the piecewise convexity of φ λ to the study of Bernoulli convolutions. Theorem 3.1. Suppose that there exists an interval (a,b) ⊂ (1,1) such that 2 φ is piecewise convex for each λ in (a,b). Then for each λ ∈ (a,b) the λ Bernoulli convolution ν is absolutely continuous with bounded density. λ 9 We stress that if φ is piecewise convex for almost every λ in (a,b), then it λ is piecewise convex for all λ in (a,b), since the maps φ are continuous in λ λ and convexity is preserved by passing to continuous limits. Proof. This theorem relies on results of Rychlik [11]. Given a function g : [0,1] → R, we define the total variation of g by n (cid:88) varg := sup |g(x )−g(x )|. i i−1 0=x0<x1<···<xn=1 i=1 The function g is said to have bounded variation if varg < ∞. Suppose that T : [0,1] → [0,1] is a piecewise continuous map, such that there exists a function g of bounded variation satisfying g = 1/|T(cid:48)| almost everywhere. We consider the transfer operator L defined on functions of bounded variation by (cid:88) Lf(x) = g(y)f(y). T(y)=x We put g = g ·(g ◦T)···(g ◦Tn−1). Then n (cid:88) Lnf(x) = g (y)f(y). n Tn(y)=x Let C denote the (n−1)th refinement of partition {[0, 1],(1,1]} by T. n 2 2 In [11], Rychlik proved that varLnf ≤ κvarf +D(cid:107)f(cid:107) , (7) 1 where κ = supg +max var g and D = max var g /|C |. n Cn Cn n Cn Cn n n We can apply this to our tent maps, replacing T with φ . Suppose that λ the tent map is convex on each of the intervals [0, 1] and [1,1]. Then φ is 2 2 λ differentiable everywhere except for at most countably many points, and this derivative is increasing on [0, 1) and on (1,1]. So there exists a function g 2 2 which is of bounded variation, which satisfies the assumptions of [11], and which satisfies g = 1 almost everywhere. We have |φ(cid:48)| λ supg = g(0) = g(1) = λ and var g = var g ≤ λ, [0,1] [1,1] 2 2 10