Docstoc

Recurrence and Hitting Problems of Continued Fraction and Beta

Document Sample
Recurrence and Hitting Problems of Continued Fraction and Beta Powered By Docstoc
					                                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

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:11
posted:3/26/2011
language:English
pages:37