Diophantine approximation of beta expansion in parameter space

Transcription

1 Diophantine approximation of beta expansion in parameter space Jun Wu Huazhong University of Science and Technology Advances on Fractals and Related Fields, France 19-25, September 2015

2 Outline 1 Background 2 β-transformation and β-expansion 3 Diophantine approximation in parameter space 4 Outline of the proof

3 1. Background

4 Background Shrinking target problem Modified Shrinking target problem

5 Shrinking target problem Suppose we have a dynamical system (X, T, µ), where µ is a T -invariant probability measure. Let A X such that µ(a) > 0. Poincaré Recurrence Theorem µ{x A : T n x A infinitely often (i.o.)} = µ(a). Birkhoff ergodic theorem Assume that µ is ergodic, then µ{x X : T n x A i.o.} = 1. That is, µ almost every x X will visit A an infinite number of times.

6 Shrinking target problem This raises the question of what happens when we allow A to shrink with respect to time i.e. let {A n } n 1 be measurable sets with µ(a n ) decreasing to 0 as n. Consider the metric properties of the following set {x X : T n x A n i.o.} This is called the Shrinking target problem or dynamical Diophantine approximation problem. The shrinking target problem was first proposed by Hill and Velani (1995) which concerns what happens if the target shrinks with the time and more generally if the target also moves around with the time.

7 Size in measure Philipp (1967) ; Boshernitzan (1993) ; Barreira and Saussol(2001) ; Chernov and Kleinbock(2001) ; Fayad(2006) ; Stefano(2007) ; Fernàndez, Meliàn and Pestana(2007, 2012) ; Saussol (2009) ; Boshernitzan and Chaika(2012, 2013) Kim and Stefano(2014)

8 Size in dimension Hill, Velani 1995, T an expanding rational map on Riemann sphere and J its Julia sets. Hill Velani (X, T ), X n-dimensional torus and T a linear operator given by a matrix with integer coefficients. Urbański, 2002 countable expanding Markov map : partial result. Stratmann and Urbański, 2002 Parabolic rational maps on Julia set. Fernàndez, Meliàn, Pestana Finite expanding Markov systems. Li-Wang-Wu-Xu, Continued fraction systems. Tan-Wang 2011, Shen-Wang, 2013, Bugeaud-Wang, Beta expansions.

9 Modified shrinking target problem Let R θ : x x + θ be a rotation map on the unit circle. Then the set studied in classical inhomogeneous Diophantine approximation can be written as {θ Q c : R n θ 0 x 0 < r n i.o. n N}. Compared with the shrinking target problem, instead of considering the Diophantine properties in one given system, here concerns the properties of the orbit of some given point in a family of dynamical systems.

10 Modified shrinking target problem Size in measure : Sprind zuk(1979), Schmidt(1980), Harman(1998), Bernik and Dodson(1999), Bugeaud(2004). Size in dimension : Levesley 1998 Bugeaud 2004, Bugeaud and Chevallier In this talk, we shall discuss Diophantine approximation properties of beta expansion in parameter space.

11 2. β-transformation and β-expansion

12 β-transformation and β-expansion Given β > 1, the β-transformation T β : [0, 1] [0, 1] is defined by T β (x) = βx (mod 1), x [0, 1]

13 There exists a unique probability measure ν β satisfying : ν β is equivalent to the Lebesgue measure L on [0, 1] with ν β is invariant under T β ; T β is ergodic with respect to ν β ; 1 1 β dν β 1 dl [0,1] 1 1 ; β ν β is the unique measure of maximal entropy.

14 β-transformation and β-expansion β-transformation was introduced by Renyi in 1957 as a model for expanding a real number in a non-integer base β > 1. The maps T β are typical examples of monotone one-dimensional expanding dynamical systems. For this class of system, it is well-known that the dynamics are determined by the orbits of the critical points. In the case of β-transformations, this critical point is the unit 1.

15 β-transformation and β-expansion digit set A = digit function ε n (x, β) := ε 1 (T n 1 β x, β) β-expansion (Rényi, 1957) digit sequence : { {0, 1,..., β 1} when β is an integer {0, 1,..., β } otherwise. x = ε 1(x, β) β ε 1 (, β) : [0, 1] A as x βx + ε 2(x, β) β ε n(x, β) β n + ε(x, β) = (ε 1 (x, β), ε 2 (x, β),..., ε n (x, β),... )

16 admissible sequence admissible sequence/word Σ β = {ω A N : x [0, 1) s.t. ε(x, β) = ω} Σ n β = {ω A n : x [0, 1) s.t. ε i (x, β) = ω i for all 1 i n} Example : β 0 = Σ β0 = {ω {0, 1} N : the word 11 dosen t appear in ω} number of admissible words of length n β n Σ n β βn+1 β 1

17 Characterization of admissible sequence the infinite expansion of the number 1 ε(1, β) if i.o. ε n (1, β) 0 ( ε (1, β) = ε1 (1, β),, (ε n (1, β) 1) ) otherwise, where εn (1, β) is the last non-zero element in ε(1, β). Theorem (Parry, 1960) Let β > 1 be a real number and ε (1, β) the infinite expansion of the number 1. Then ω Σ β if and only if σ k (ω) ε (1, β) for all k 0, where means the lexicographical order.

18 Self-admissible sequence Corollary (Parry, 1960) w is infinite β-expansion of 1 for some β σ k (w) w for all k 0. self-admissible word w = (ε 1, ε 2, ) σ k w w for all k 1. Note : NOT all sequences are the expansion of 1 under some β. The interested sequences have to be chosen carefully.

19 3. Diophantine approximation in parameter space

20 Diophantine approximation in parameter space Given a point x (0, 1], its orbits under T β may have completely different distributions on [0, 1] when β varies. The parameter space {β R: β > 1} can be divided into five classes according to the distributions of the orbits of 1, see Blanchard, C 1 : O β is ultimately zero. O β := {T n β 1: n 1} C 2 : O β is ultimately non-zero periodic. C 3 : O β is an infinite set but 0 is not an accumulation point of O β. C 4 : 0 is an accumulation point of O β but O β is not dense in [0, 1]. C 5 : O β is dense in [0, 1].

21 Diophantine approximation of the orbit of 1 in parameter space Size in measure (Schmeling, 1997) : for almost all β > 1, {T n β 1} n 1 is dense. i.e., for any x 0 [0, 1], lim inf n T n β 1 x 0 = 0 for L-a.e. β > 1. For any x 0 [0, 1] and any sequence of positive real numbers {l n } n 1, set E(x 0, {l n }) := { } β > 1 : Tβ n 1 x 0 < β ln, i.o. n N Size in dimension When x 0 = 0 and l n = αn(α > 0), Persson and Schmeling (2008) proved that dim H E(0, {αn}) = α.

22 Diophantine approximation of the orbit of 1 in parameter space Theorem (Li, Persson, Wang and W, 2014) Let x 0 [0, 1] and {l n } be a sequence of positive real numbers such that l n as n. Then dim H E(x 0, {l n }) = 1 l n, where α = lim inf 1 + α n n. Theorem (Li, Persson, Wang and W, 2014) Suppose x 0 = x 0 (β): (1, + ) [0, 1] is a Lipschitz continuous function and {l n } is a sequence of positive real numbers such that l n as n. Then dim H { β > 1: T n β 1 x 0 < β ln i.o. n N } = α, where α = lim inf n l nn.

23 Diophantine approximation of the orbit of 1 in parameter space The main difficulties are A general idea to make sure T n β 1 x 0 small is that let (ɛ n+1 (1, β), ɛ n+2 (1, β), ) and ɛ(x 0, β) have sufficiently long common prefix. But the problem is that since β has not been fixed, we do not know what the expansion of x 0 should be. We introduce a notion, called recurrence time of a word, to characterize the lengths and the distribution of cylinders in the parameter space {β R : β > 1}. The above results only consider the Diophantine approximation properties of the orbit of 1, not for general x (0, 1].

24 Diophantine approximation of the orbit of general point in parameter space Fix x (0, 1]. Size in measure (Schmeling, 1997) : for almost all β > 1, {T n β x} n 1 is dense. i.e., for any x 0 [0, 1], lim inf n T n β x x 0 = 0 for L-a.e. β > 1. Let x 0 [0, 1] and ϕ: N (0, 1] be a positive function, define Ẽ(x 0, ϕ) := {β > 1: T n β x x 0 < ϕ(n) i.o. n N} Let x 0 [0, 1] and {l n } n 1 be a sequence of positive real numbers, define Ẽ(x 0, {l n }) := { } β > 1 : Tβ n x x 0 < β ln, i.o. n N.

25 Main results Theorem (Lü and W, 2015) dim H Ẽ(x 0, ϕ) = Theorem (Lü and W, 2015) where α = lim inf n l nn. 0, if lim sup n 1, if lim sup n log ϕ(n) n = ; log ϕ(n) n dim H Ẽ(x 0, {l n }) = α, >.

26 4. Outline of the proof

27 admissible word From now on, x is always a fixed point in (0, 1]. Definition We say w is an admissible word of length k if there exists some β > 1 such that ε 1 (x, β) ε k (x, β) = w. Let Ω k = {admissible word of length k} and Ω = k=1 Ω k. For any w Ω and k 1, denote by w (k) the lexicographically largest admissible word of length w + k such that w is a prefix of w (k). Let w ( ) be the infinite sequence of nonnegative integers such that w (k) is a prefix of w ( ) for all k 1.

28 admissible word For any k 1 and w Ω k, let β(w) = 1 if w = 0 k ; otherwise, let β(w) 1 be the unique positive solution of the equation x = k i=1 i=1 w i β i. For any n 1, let β(w (n) ) 1 be the unique positive solution of the equation k+n w (n) i x = β i. Lemma For any k 1 and w Ω k, the limit of the sequence { β ( w (n))} exists. If we denote the limit by β(w), then w ( ) = w 1 w i ε ( 1, β(w) ) for some 1 i k.

29 cylinders in the parameter space For any k 1 and w Ω k, let I(w) = {β > 1: ε 1 (x, β) ε k (x, β) = w}, i.e the collection of β such that the β-expansion of x begins with w. Lemma Given k 1 and w Ω k, if β(w) > 1, then the set I(w) is a half open interval [ β(w), β(w) ) ; if β(w) = 1, then the set I(w) is an open interval ( 1, β(w) ).

30 perfect words Given k 1, a word w Ω k is said to be perfect if J(w) := { T k β x: β I(w) } = [0, 1). If w is a perfect word, we call I(w) a regular cylinder. Theorem For any k 1, there exists at least one perfect word in k + 1 consecutive words in Ω k. This tells us that regular cylinders are sufficiently well distributed. So, as far as Hausdorff dimension is concerned, it should be OK by focusing only on regular cylinders.

31 A criterion of regular cylinder A criterion to find regular cylinders. Lemma Let (ε 1,, ε k 1, ε k ) and (ε 1,, ε k 1, ε k ) be two words. If ε k < ε k and (ε 1,, ε k 1, ε k ) Ω k, then (ε 1,, ε k 1, ε k ) Ω k and (ε 1,, ε k 1, ε k ) is a perfect word.

32 perfect words Theorem For any k 1 and w Ω k, we have I(w) ( β(w) ) 1 k. Furthermore, if the word w is perfect, then I(w) β(w) (β(w) 1) 2 ( β(w) ) 1 k.

33 Theorem For any k 1 and w Ω k, the set I(w; ϕ) := { β I(w): Tβ k x x 0 < ϕ(k) } is an interval with I(w; ϕ) 2x 1 ϕ(k) ( β(w) ) 1 k. Furthermore, if the word w is perfect, then I(w; ϕ) (kx) 1 ϕ(k) ( β(w) ) 1 k.

34 Mass transference principle Mass transference principle was established by Beresnevich and Velani in It allows us to tranfer Lebesgue measure theoretic statements for limsup sets to the Hausdorff measure theoretic statements. It has become a very important tools to estimate the Hausdorff measure and Hausdorff dimension of limsup sets. Theorem (Beresnevich and Velani, 2006) Let X be a closed interval and let {B(y j, r j )} be a sequence of balls on R with r j 0 as j. Given s (0, 1), suppose that for any ball B in X ) H (B 1 lim sup B(y j, rj) s = H 1 (B). j Then, for any ball B in X ) H (B s lim sup B(y j, r j ) j = H s (B).

35 Lower bound For any 1 < β 1 < β 2, let F x (β 1, β 2 ) = { β [β 1, β 2 ]: T n β x x 0 < ϕ(n) i.o. n N }. we prove that dim H F x (β 1, β 2 ) s(β 1, β 2 ) := log β 1 log β 2 lim sup n. log ϕ(n) n For any s < s(β 1, β 2 ), choose a subsequence {n k } of positive integers such that s + s(β 1, β 2 ) log β 1 < 2 log β 2 lim sup k. log ϕ(n k ) n k

36 Lower bound For any k 1, we can find m k perfect words satisfying ζ nk,1 ζ nk,2 ζ nk,m k ; ζ nk,1, ζ nk,2,, ζ nk,m k Ω nk I(ζ nk,i) [β 1, β 2 ] for all 1 i m k ; For any z nk,i I(ζ nk,i), 1 i m k, [β 1, β 2 ] m k i=1 B ( z nk,i, (n k + 2)x 1 β 1 n ) k 1.

37 Lower bound For any k 1 and 1 i m k, the set I(ζ nk,i) contains an interval I(ζ nk,i; ϕ) with Choose I(ζ nk,i; ϕ) (2n k x) 1 ϕ(n k )β 1 n k 2. A nk,i := B ( z nk,i, (4n k x) 1 ϕ(n k )β 1 n ) k 2 I(ζnk,i; ϕ) Denote the sequence A n1,1, A n1,2,, A n1,m 1, A n2,1, A n2,2,, A n2,m 2, A n3,1, by B(y 1, r 1 ), B(y 2, r 2 ), B(y 3, r 3 ),.

38 Lower bound One can check that r j 0 as j ; lim sup B ( y j, rj) s [β1, β 2 ]. j Then by the mass transference principle, we have ) H s (F x (β 1, β 2 )) H ([β s 1, β 2 ] lim sup B(y j, r j ) j = H s ([β 1, β 2 ]) =.

39 Thanks for your attention!