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Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space Diophantine approximation of the orbit of 1 in beta-transformation dynamical system Bing LI (joint work with Baowei Wang and Jun Wu) South China University of Technology and University of Oulu CUHK, December, 2012 International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space Outline Diophantine approximation of the orbits of 1 under 1 beta-transformations β -transformation and β -expansion 2 Distribution of regular cylinders in parameter space 3 International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space Diophantine approximation of the orbits of 1 under beta-transformations International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space Backgrounds Poincar´ e Recurrence Theorem Let ( X, B , µ, T ) be a measure-preserving dynamical system (probability space) and B ⊂ X with positive measure. Then µ { x ∈ B : T n x ∈ B infinitely often (i.o.) } = µ ( B ) . Birkhoff ergodic theorem Assume that µ is ergodic, then µ { x ∈ X : T n x ∈ B i.o. } = 1 . dynamical Borel-Cantelli Lemma or shrinking target problem Let { B n } n ≥ 1 be a sequence of measurable sets with µ ( B n ) decreasing to 0 as n → ∞ . Consider the metric properties of the following set { x ∈ X : T n x ∈ B n i.o. } International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space Backgrounds well-approximable set Let d be a metric on X consistent with the probability space ( X, B , µ ) . Given a sequence of balls B ( y 0 , r n ) with center y 0 ∈ X and shrinking radius { r n } , the set F ( y 0 , { r n } ) := { x ∈ X : d ( T n x, y 0 ) < r n i.o. } is called the well-approximable set. inhomogeneous Diophantine approximation Let S α : x �→ x + α be the irrational rotation map on the circle with α / ∈ Q . The classic inhomogeneous Diophantine approximation can be written as α ∈ Q c : | S n � � α 0 − y 0 | < r n , i.o. n ∈ N . International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space beta-transformations β > 1 β -transformation T β : [0 , 1] → [0 , 1] T β ( x ) = βx − ⌊ βx ⌋ , where ⌊ βx ⌋ denotes the integer part of βx . √ Example : β = 1+ 5 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 the orbit of 1 under T β is crucial (we will see later) International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space Main problem well-approximable set Fix a point x 0 ∈ [0 , 1] and a given sequence of integers { ℓ n } n ≥ 1 . β > 1 : | T n β 1 − x 0 | < β − ℓ n , i.o. � � � � E { ℓ n } n ≥ 1 , x 0 = Question : � � dim H E { ℓ n } n ≥ 1 , x 0 =? (Persson and Schmeling, 2008) When x 0 = 0 and ℓ n = γn ( γ > 0) , then 1 dim H E ( { γn } n ≥ 1 , 0) = 1 + γ . International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space Main result Theorem Let x 0 ∈ [0 , 1] and let { ℓ n } n ≥ 1 be a sequence of integers such that ℓ n → ∞ as n → ∞ . Then 1 ℓ n � � dim H E { ℓ n } n ≥ 1 , x 0 = 1 + α, where α = lim inf n . n →∞ International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space β -transformation and β -expansion International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space Recall beta-transformations β > 1 β -transformation T β : [0 , 1] → [0 , 1] T β ( x ) = βx − ⌊ βx ⌋ , where ⌊ βx ⌋ denotes the integer part of βx . √ Example : β = 1+ 5 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space Invariant measure (R´ enyi 1957) When β is not an integer, there exists a unique invariant measure µ β which is equivalent to the Lebesgue measure. 1 − 1 β ≤ dµ β 1 d L ( x ) ≤ 1 − 1 β Equivalent invariant measure µ β (Parry 1960 and Gel’fond 1959) dµ β 1 1 � d L ( x ) = F ( β ) β n n ≥ 0 x<T n β (1) � 1 � β (1) 1 /β n dx is a normalizing factor. where F ( β ) = n ≥ 0 x<T n 0 International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space β -expansion digit set � { 0 , 1 , . . . , β − 1 } when β is an integer A = { 0 , 1 , . . . , ⌊ β ⌋} otherwise. digit function ε 1 ( · , β ) : [0 , 1] → A as x �→ ⌊ βx ⌋ ε n ( x, β ) := ε 1 ( T n − 1 x, β ) β β -expansion (R´ enyi, 1957) x = ε 1 ( x, β ) + ε 2 ( x, β ) + · · · + ε n ( x, β ) + · · · β 2 β β n notation : ε ( x, β ) = ( ε 1 ( x, β ) , ε 2 ( x, β ) , . . . , ε n ( x, β ) , . . . ) International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space admissible sequence admissible sequence/word Σ β = { ω ∈ A N : ∃ x ∈ [0 , 1) such that ε ( x, β ) = ω } β = { ω ∈ A n : ∃ x ∈ [0 , 1) such that ε i ( x, β ) = ω i for all i = 1 , · · · , n } Σ n β is an integer Σ β = A N ( except countable points ) √ 5+1 Example : β 0 = 2 Σ β 0 = { ω ∈ { 0 , 1 } N : the word 11 dosen’t appear in ω } number of admissible words of length n β ≤ β n +1 β n ≤ ♯ Σ n β − 1 International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space admissible sequence the infinite expansion of the number 1  ε (1 , β ) if there are infinite many    ε n (1 , β ) � = 0 in ε (1 , β )     � ∞ ε ∗ (1 , β ) = � ε 1 (1 , β ) , · · · , ( ε n (1 , β ) − 1) otherwise, where ε n (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. International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical

Diophantine approximation of the orbits of 1 under beta-transformations β -transformation and β -expansion Distribution of regular cylinders in parameter space self-admissible sequence Corollary (Parry, 1960) ⇒ σ k ( w ) � w for all k ≥ 0 w is the β -expansion of 1 for some β ⇐ self-admissible sequence σ k ( w ) � w for all k ≥ 0 International conference on advances on fractals and related topics Diophantine approximation of the orbit of 1 in beta-transformation dynamical