# Recurrence and Hitting Problems of Continued Fraction and Beta

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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
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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
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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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