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Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem Recurrence and Hitting Problems of Continued Fraction and Beta-Shift Dynamical Systems Bing LI Department of Mathematics, National Taiwan University NCTS/TPE HsinChu, May 15, 2010 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem Plan 1 Continued fraction dynamical system 2 β-shift dynamical system 3 Compare between the continued fraction and β-shift dynamical system 4 Recurrence problem 5 Hitting problem 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem Continued fraction dynamical system 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem I. Continued fraction dynamical system ([0, 1), B, L, TG ) Probability space : ([0, 1), B, L) 1 Gauss transformation : TG (x) = x ( mod 1) if x = 0, TG (x) = 0 if x=0 The transformation TG is not measure-preserving w.r.t. L. 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem II. Continued fraction expansion Invariant measure The Gauss measure µG deﬁned by 1 1 dµG (x) = dx log 2 1 + x is TG -invariant and equivalent to the Lebesgue measure. Continued fraction expansion x := [a1 (x), a2 (x), . . .] = 1 1 a1 (x) + . a2 (x) + . . a1 (x), a2 (x), . . . are called the partial quotients of the continued fraction expansion of x. Algorithm 1 a1 (x) = [ x ] a2 (x) = a1 (TG x) . . . n−1 an (x) = a1 (TG x) . . 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem III. Examples of continued fractions Examples : √ 5+1 = [1; 1, 1, . . . ] = [1; 1]. √2 √ 2 = [1; 2, 2, . . . ] = [1; 2] 3 = [1; 1, 2, 1, 2, . . . ] = [1; 1, 2] π = [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, , . . . ] Remark : The number x is quadratic ⇐⇒ its continued fractions is ultimately periodic. The number x is rational ⇐⇒ its continued fractions is ﬁnite. 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem IV. Conjugate to symbolic space Projective map π : [0, 1) → π(x) = (a1 (x), a2 (x), . . . ) = {1, 2, . . . , n, . . .}N and σ : → shift transformation π : one-to-one Commutative diagram TG [0, 1) - [0, 1) π π ? ? - σ π ◦ TG = σ ◦ π µ = µG ◦ π −1 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem IV. Conjugate to symbolic space Isomorphism ([0, 1), B, µG , TG ) ∼ ( , B, µ, σ) π is not continuous ([0, 1), TG ) is not topological conjugate to ( , σ) 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem V. Topological entropy and measure-theoretical entropy Topological entropy ∞ Measure-theoretical entropy π2 hµG (TG ) = 6 log 2 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem β-shift dynamical system 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem I. β-shift dynamical system([0, 1), B, L, Tβ ) Let β > 1 be any real number, not an integer. Probability space : ([0, 1), B, L) β-transformation : Tβ x = {βx} The transformation Tβ is not measure-preserving. √ Example : β = 1+2 5 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem II. Invariant measure −1 a 1 1 Fact : Tβ [0, a) = [0, β ) ∪ [ β , 1) β ≤ a ≤ 1 −1 a 1 1 Tβ [0, a) = [0, β ) ∪ [ β , 1+a ) 0 ≤ a < β β The unique Tβ -invariant measure equivalent to L is deﬁned as dµβ 1 = F (β) dx n βn x<Tβ (1) 1 where F (β) = 0 x<T n (1) 1/β n is a normalizing factor. ( A. O. β Gel’fond 1959 and W. Parry 1960) 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem III. β-expansion Let β > 1 be any real number and not an integer. e β-expansion (A. R´nyi, 1957) ε1 (x) ε2 (x) εn (x) x= + + ··· + + ··· (1) β β2 βn where ε1 (x), ε2 (x), . . . , εn (x), . . . ∈ {0, 1, . . . , [β]}, are called the β-digits of x and (1) is called the β-expansion of x in base β, denote by (ε1 (x), ε2 (x), . . . , εn (x), . . .). algorithm (greedy algorithm) ε1 (x) = [βx] ε2 (x) = ε1 (Tβ x) . . . n−1 εn (x) = ε1 (Tβ x) · · · 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem IV. Speed of convergent ε1 (x) ε2 (x) εn (x) ωn (x) = β + β2 + ··· + βn ωn (x) → x as n → ∞ Problem : what is the convergence speed of ωn (x) for x ∈ [0, 1) ? Theorem (Fan and L., preprint) Let β > 1 be a real number. We have 1 lim logβ (x − ωn (x)) = −1 n→∞ n for almost all x ∈ [0, 1) w.r.t. the Lebesgue measure. 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem V. Fundamental intervals the β-digits εn (x) ∈ A = {0, 1, · · · , [β] − 1} when β is an integer and A = {0, 1, · · · , [β]} when β is not an integer. n-th cylinder : J(ε1 , ε2 , · · · , εn ) = {x ∈ [0, 1) : εk (x) = εk , 1 ≤ k ≤ n} Remark : The cylinder J(ε1 , ε2 , · · · , εn ) may be empty ! Admissible sequence ω = (ε1 , · · · , εn , · · · ) : if ∃x ∈ [0, 1] such that the β-expansion of x is just ω Dβ = {ω ∈ A : ω is an admissible sequence} J(ε1 , · · · , εn ) = ∅ ⇔ (ε1 , · · · , εn , 0∞ ) ∈ Dβ 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem VI. Admisssible sequence Dβ =? ε(1, β) : the β-expansion of the number 1 √ Example : β0 = 5+1 , then W = {0, 1}, 2 Dβ0 = {ω ∈ W : the word 11 dosen’t appear in ω} ε(1, β0 ) = (1, 1, 0, . . . , 0 . . . ) Remark : ε(1, β) plays a very important role in the characterization of Dβ . 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem → If there are inﬁnitely many non-zero elements in the sequence ε(1, β), we said that ε(1, β) is inﬁnite, otherwise, it is said to be ﬁnite. In the ﬁnite case, i.e., ε(1, β) = ε1 (1), · · · , εn (1), 0∞ with εn (1) = 0, ∞ ω ∈ Dβ , we take ε∗ (1, β) = ε1 (1), ε2 (1), · · · , εn−1 (1), (εn (1) − 1) as the inﬁnite expansion of 1. In the following, we will write ε(1, β) instead of ε∗ (1, β) for the ﬁnite case. Theorem (W. Parry, 1960) Let β > 1 be a real number. Then ω ∈ Dβ if and only if σ k (ω) <lex ε(1, β) for all k ≥ 0. the lexicographical ordering <lex on W : (ε1 , ε2 , · · · , εn , · · · ) <lex (ε1 , ε2 , · · · , εn , · · · ) means that there exists k ≥ 1 such that εj = εj for all 1 ≤ j < k and εk < εk 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem VII. Admisssible sequence Theorem (W. Parry, 1960) A sequence ω = {ωn } ∈ {0, 1, . . . , N }N is an expansion of the number 1 for some β if and only if σ k (ω) <lex ω(k > 0) and then β is unique. The map τ : β → ε(1, β) is monotone increasing. 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem VIII. Project to symbolic dynamics Projective map π : [0, 1) → Dβ π(x) = (ε1 (x), ε2 (x), . . . ) π : one-to-one, right-continuous,strictly increasing π −1 : continuous, strictly increasing, onto Commutative diagram Tβ [0, 1) - [0, 1) π π ? ? Dβ - Dβ σ π ◦ Tβ = σ ◦ π ν = µβ ◦ π −1 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem IX. Measure-theoretic and topological entropy subshift (Sβ , σ) Sβ is the closure of Dβ Proposition ( S. Ito and Y. Takahashi, 1974) 1. 1 < β ≤ α =⇒ Dβ ⊂ Dα 2.∀β > 1, Sβ = α<β Dα = α>β Dα conjugate ([0, 1), B, µβ , Tβ ) ∼ (Sβ , B, ν, σ) measure-theoretic entropy for µβ hµβ (Tβ ) = hν (σ) = log β maximal entropy measure The measure µβ is the unique measure of maximal entropy. (F. Hofbauer, 1978) e topological entropy(R´nyi, 1956, S. Ito and Y. Takahashi, 1974) h(Tβ ) = h(σ|Sβ ) = log β 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem IV. A classiﬁcation of all integers greater than 1 Notation : ln = sup{k ≥ 0 : ε∗ (1) = 0 for all 1 ≤ j ≤ k} n+j Classiﬁcation ( L. and Wu 2008) A0 = {β ∈ (1, +∞) : ln = O(1)} A1 = {β ∈ (1, +∞) : ln = o(n), ln = O(1)} A2 = {β ∈ (1, +∞) : ln = o(n)} Remark : A0 A1 A2 = (1, +∞) {Parry numbers} ⊂ A0 L(A0 ) = 0, dimH (A0 ) = 1 It is diﬀerent with the classiﬁcation of F. Blanchard (1989). 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem V. Lengthes of cylinders n-th cylinder : J(ε1 , · · · , εn ) = {x ∈ [0, 1) : ε1 (x) = ε1 , · · · , εn (x) = εn } 1 N -adic expansion : the lengthes of all n-th cylinders is equal to Nn . Theorem (L. and Wu, 2008) 1 1 β ∈ A0 ⇐⇒ C ≤ J(ε1 (x), · · · , εn (x)) ≤ n ∀x ∀n, βn β where C is a constant. Theorem (L. and Wu, 2008) log |J(ε1 (x), ε2 (x), · · · , εn (x))| β ∈ A0 ∪ A1 ⇐⇒ lim − = log β ∀x n→∞ n 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem Compare between the continued fraction and β-shift dynamical system 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem Compare between the continued fraction and β-shift dynamical system Properties β-shift Continued fraction topological entropy log β ∞ invariant measure µ (equi- Parry measure Gauss measure valent to L) π2 measure-theoretic entropy log β 6 log 2 (w.r.t. µ) Symbolic space [β] + 1 countable Mixing exponential mixing exponential mixing 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem Recurrence problem 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem e I. Poincar´ recurrence theorem Let (X, B, µ, T ) be a measure-preserving dynamical system (µ ◦ T −1 = µ). e e Poincar´ Recurrence Theorem (Poincar´, 1899) Let B ∈ B. For almost all point x ∈ B, the orbits {T n x}n≥1 return to B inﬁnite often. Recurrent Let (X, d) be a metric space compatible with B. The point x is said to be recurrent if lim inf d(T n x, x) = 0. n→∞ Corollary : Assume that (X, d) has a countable base, µ-almost all x ∈ X is recurrent. If, furthermore, µ is ergodic, then µ-almost all x ∈ X hit every y ∈ X in the sense lim inf d(T n x, y) = 0. n→∞ 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem II. Quantitative recurrence theorem Quantitative recurrence theorem (Boshernitzan, 1993) Let (X, B, µ, T, d) be a metric measure preserving system. Assume that, for some α > 0, the Hausdorﬀ α-measure H α is σ-ﬁnite on X. Then for µ-almost all x ∈ X, we have 1 lim inf n α d(T n x, x) < ∞. n→∞ If, moveover, H α (X) = 0, then for µ-almost all x ∈ X, 1 lim inf n→∞ n α d(T n x, x) = 0. Applications to these two dynamics Let ([0, 1), B, L, T, d) be continued fraction or β-shift dynamical systems. Then for µ-almost all x ∈ X, we have lim inf n|T n x − x| < ∞. n→∞ 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem III. Dimensional result for continued fraction dynamics Let φ : N → R+ with φ(n) → ∞ as n → ∞. Set F (φ) = x ∈ [0, 1) : T n (x) − x < (φ(n))−1 , i.o. n ∈ N . Theorem (L., Wang, Wu and Xu, preprint) log φ(n) Write lim inf n = log B. n→∞ (1) If B = 1, then dimH F (φ) = 1. (2) If 1 < B < ∞, then dimH F (φ) = sB , where sB is the solution to the equation P (−s log |T (x)|) = s log B. log log φ(n) (3) If B = ∞, write lim inf n = log b. n→∞ 1 (3a) If b = 1, then dimH F (φ) = 2 . 1 (3b) If 1 < b < ∞, then dimH F (φ) = 1+b . (3c) If b = ∞, then dimH F (φ) = 0. 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem Hitting problem 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem I. Fixed target problem Let (X, B, µ) be a measure-preserving dynamical system and {rn } a decreasing sequence of positive numbers. Let x0 be a ﬁxed point of X. Shrinking targets : {B(x0 , rn )} Shrinking target problem : For x ∈ X, how many times do the orbit of x hit the shrinking target in the ﬁrst n times ? a a (Fern´ndez, Meli´n and Pestana, 2007) Let ([0, 1), B, µ, T, d) be continued fraction or β-shift dynamical systems. Then n i i=1 χB(x0 ,ri ) (T x) lim n =1 i=1 µ(B(x0 , rn )) n→∞ for µ-almost every x ∈ X. 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem II. Moving target problem For x ∈ X, how many times at most or at least during the ﬁrst N times do the orbit of x hit every shrinking target, that is to say, N N min χB(y,rn ) (T n x) =? max χB(y,rn ) (T n x) =? y∈X y∈X n=1 n=1 Theorem (Fan, Langlet and L. preprint) log2 n If lim = 0, then for µ-almost all x ∈ X, n→∞ nrn N N min n=1 χB(y,rn ) (T n x) max n=1 χB(y,rn ) (T n x) y∈X y∈X 0 < lim inf N ≤ lim sup N <∞ N →∞ n=1 rn N →∞ n=1 rn 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem III. Uniformly hitting rate Hitting rate problem : For which τ , we have lim inf nτ d(T n x, y) < ∞ (∀y ∈ X)? n→∞ Theorem (Fan, Langlet and L. preprint) Let ([0, 1), B, µ, T, d) be continued fraction or β-shift dynamical systems. If τ < 1, then for µ-almost all x ∈ X, we have lim inf nτ d(T n x, y) < 1 (∀y ∈ X). n→∞ If τ > 1, then there exists y ∈ X such that lim nτ d(T n x, y) = ∞ (a. e. x ∈ X). n→∞ 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem IV. General results Assumptions (ξn )n≥0 is a stationary process deﬁned on a probability space (Ω, A , P), taking values in a metric space (X, d). The measure µ is the initial probability measure deﬁned by µ(B) = P (ξ0 ∈ B) for any Borel set B in X. There exist two increasing functions ϕ1 and ϕ2 such that ϕ1 (r) ≤ µ(B(x, r)) ≤ ϕ2 (r) (∀x ∈ X, ∀r > 0) (2) where ϕ1 and ϕ2 satisfy limr→0 ϕi (r) = 0 and ϕi (2r) ≤ Cϕi (r) i = 1, 2, ∀r > 0. The sequence of positive number {rn } satisﬁes log2 n lim = 0. (3) n→∞ nϕ1 (rn ) 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem IV. General result Exponentially mixing The process (ξn ) is said to be exponentially mixing if there exist two constants c > 0 and 0 < γ < 1 such that |P(ξ0 ∈ E|D) − P(ξ0 ∈ E)| ≤ cγ n (4) holds for any n ≥ 1, for any ball E in X and any D ∈ An . If the condition (4) still holds for any measurable set E, it describes the φ-mixing with the exponential decay. an i.i.d. sequence is exponentially mixing exponentially mixing dynamical system : the process (ξn ) deﬁned by ξn (x) = T n x is an exponentially mixing process Examples : continued fraction transformation with Gauss measure, β-transformation with Parry measure. 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem IV. General result Covering number Theorem (Fan, Langlet and L. preprint) Suppose (ξn ) is an exponentially mixing stationary process and its initial probability satisﬁes the assumption (2). Suppose (rn ) is a decreasing se- quence of positive numbers satisfying the condition (3). Then N min n=1 χB(y,rn ) (ξn (ω)) y∈X a.s. lim inf N > 0. N →∞ ϕ1 (rn ) n=1 and N max n=1 χB(y,rn ) (ξn (ω)) y∈X a.s. lim sup N < +∞. N →∞ n=1 ϕ2 (rn ) 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem IV. General result Hitting rate log µ(B(x,r)) α∗ = supx∈X lim inf r→0 log r αmax = lim supr→0 supx∈X log µ(B(x,r)) log r 1 1 αmax ≤ α ∗ Theorem (Fan, Langlet and L. preprint) Let (X, B, µ, T, d) be an m.m.p.s. Suppose that the system is exponentially mixing and αmax < ∞. If τ < 1/αmax , then for µ-almost all x ∈ X, we have lim inf nτ d(T n x, y) = 0 (∀y ∈ X). n→∞ ∗ If τ > 1/α , then there exists y ∈ X such that lim nτ d(T n x, y) = ∞ (a.e. x ∈ X). n→∞ 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems Continued fraction dynamical system β-shift dynamical system Compare between the continued fraction and β-shift dynamical system Recurrence problem Hitting problem Thanks for your attention ! 2010 NCTS Workshop on Dynamical Systems Recurrence and Hitting Problems

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