The multifractal spectra for the recurrence rates of
beta-transformations
Jung-Chao Ban and Bing Li
∗November 16, 2011
Abstract
In this paper, the multifractal spectra for the recurrence rates of the first return time of β-transformations are studied for any β > 1, including cases with specification property and beyond. Let A(log τ
β n(x)
n ) be the set of all accumulation
points of the first recurrence rate of x∈ [0, 1). We found that the set of points satisfying A(log τ
β n(x)
n ) equals any given closed interval of the half real line is of
full Hausdorff dimension. As a consequence, the level sets concerning such rates have the constant spectra.
Key Words: recurrence, β-expansion, multifractal, Hausdorff dimension AMS Subject Classification: 37E05, 11A63
1
Introduction
Let (X,B, µ, T, d) be a metric measure-preserving system (m.m.p.s.), by which we mean that (X, d) is a metric space, B is a σ-field containing the Borel σ-field of X and (X,B, µ, T ) is a measure-preserving dynamical system. Under the assumption that (X, d) has a countable base, Poincar´e recurrence theorem implies that µ-almost all
x∈ X is recurrent in the sense
lim inf
n→∞ d(T n
x, x) = 0 (1.1)
(for example, see [11]). Later, Boshernitzan [5] has improved it by a quantitative result
lim inf
n→∞ n
1/αd(Tnx, x) <∞, µ-almost everywhere (a.e. for short),
where α is the dimension of the space in some sense.
The above results describe whether or not a point is recurrent and how far the orbit will return to the initial point. Recurrence time is an important aspect used to characterize the behaviors of orbits in dynamical systems. Of the research conducted on recurrence time, the first return time of a point has been well studied in the last decade. The first return time of a point x∈ X into the set A is defined by
τA(x) = inf{k ∈ N : Tkx∈ A}.
Ornstein and Weiss [21] proved that for a finite partition ξ of X, if there exists a
T -invariant ergodic Borel probability measure µ, then
lim
n→∞
log τξn(x)(x)
n = hµ(ξ), µ− a.e.
where ξn(x) is the intersection of ξ, T−1(ξ), · · · , T−n+1(ξ) which contains x, and hµ(ξ)
denoted by the measure-theoretic entropy of T with respect to the partition ξ. Feng and Wu [10] considered the recurrence set of the one-sided shift space on m symbols ({0, 1, . . . m − 1}N, σ), where the partition ξ is the cylinders sets {[0], [1], . . . , [m − 1]}.
They proved that the set { x∈ {0, 1, . . . m − 1}N: lim inf n→∞ log τξn(x)(x) n = α, lim supn→∞ log τξn(x)(x) n = γ }
has Hausdorff dimension one for any 0 ≤ α ≤ γ ≤ +∞ (see also [27]). Lau and Shu [15] extended this result to the dynamical systems with specification property by considering the topological entropy instead of Hausdorff dimension. Barreira and Saussol [1] replaced the cylinders ξn(x) with the balls B(x, r) according to quantity
τr(x) = inf{n ≥ 1 : Tnx∈ B(x, r)},
and define the lower and upper recurrence rate of x by
R(x) = lim inf r→0
log τr(x)
− log r , R(x) = lim supr→0
log τr(x) − log r .
They proved that
R(x) = dµ(x), R(x) = dµ(x), µ− a.e. (1.2)
with the conditions that µ has a so-called long return time (see [1]) and dµ(x) > 0 for
µ-a.e. x, where dµ(x), dµ(x) are the lower and upper pointwise dimensions of µ at a
point x ∈ X respectively. A simple consequence of this result is a reformulation of Boshernitzan’s theory by noting that
lim inf
n→∞ n
holds for all α > dµ(x). Many researchers (for example, [2, 26, 28]) have studied the problem when the formulation (1.2) holds from many different viewpoints. For example, Barreira and Saussol [2] proved that formulation (1.2) holds for several differ-ent situations, including repellers of C1+α expanding maps together with equilibrium measures of H¨older continuous functions and the continued fraction transformation endowed Gauss measure.
Let A(log τr(x)
− log r ) be the set of the accumulation points of
log τr(x)
− log r as r → 0 and J a
compact sub-interval of (0, +∞). Olsen [20] studied the following set
G∩ {x ∈ K : A ( log τr(x) − log r ) = J}
for the self-conformal set (satisfying a certain separation condition) K with the natural self-map induced by the shift, where G is an open set with G∩ K ̸= ∅. He proved that such a set shares the same Hausdorff dimension as K. This result can be applied to the case of N -adic transformation with N ∈ N.
In this investigation we consider the similar problem for the β-transformation with any β > 1, which includes the cases of full-shift (β = N ), subshift of finite type, and cases with, but not limited to, specification condition.
For any n∈ N and x ∈ [0, 1), define
τnβ(x) = inf{m ≥ 1 : Tβmx∈ In(x)},
where Tβ is the β-transformation and In(x) is the n-th cylinder containing x. Apply the
metric result in [2] and Shannon-McMillan-Breiman Theorem (see [7] Theorem 6.2.1), note that dµ
β(x) = dµβ(x) = 1 forL-almost every x ∈ [0, 1) and the measure-theoretical
entropy of Tβ with respect to the Parry measure µβ is log β, as such we know that
lim n→∞ log τβ n(x) n = limn→∞ log τβ n(x) − log |In(x)|nlim→∞
− log |In(x)|
n = log β
for L-almost every x ∈ [0, 1). Denoted by A (
log τnβ(x)
n
)
the set of all accumulation
points of log τnβ(x)
n as n→ ∞. The following is the main result of this paper.
Theorem 1.1. Let β > 1 be any real number and J a closed interval in (0, +∞).
Denote EJβ = { x∈ [0, 1) : A ( log τβ n(x) n ) = J } . Then dimHEJβ = 1,
Based on the corollaries of this theorem, we obtained some dimensional results for the level sets.
Corollary 1. Let β > 1 be a real number and 0 ≤ α ≤ γ ≤ +∞. Denote
Eα,γβ = { x∈ [0, 1) : lim inf n→∞ log τnβ(x) n = α, lim supn→∞ log τnβ(x) n = γ } . Then dimHEα,γβ = 1. Proof. Since E[α,γ]β = { x∈ [0, 1) : A ( log τβ n(x) n ) = [α, γ] } ⊂ Eβ α,γ,
applying Theorem 1.1 to the set E[α,γ]β , we obtain that dimHEα,γβ = 1.
Choose α = γ in Corollary 1, we have the following.
Corollary 2. Let β > 1 be a real number and 0 ≤ α ≤ +∞. Denote
Eαβ = { x∈ [0, 1) : lim n→∞ log τβ n(x) n = α } . Then dimHEαβ = 1.
The paper is organized as follows. Definitions and known results of β-transformations, as well as Hausdorff dimensions and measures, are given in Section 2. Followed by a detailed proof for Theorem 1.1 in Section 3.
2
Preliminaries
2.1
Basic notions and notation for β-transformations
R´enyi [25] introduced the β-expansions of real numbers in 1957, where 1 < β ∈ R. More specifically stated, the β-expansion of x ∈ [0, 1) is the following
x = ∞ ∑ n=1 εn(x, β) βn , (2.3)
where ε1(x, β) = [βx], [x] is the integer part of x and εn(x, β) = ε1(Tβn−1(x), β) for all n ≥ 2. Here Tβ is the β-transformation on the unit interval [0, 1) defined as
The numbers ε1(x, β), ε2(x, β), . . . , εn(x, β), . . . are the β-digits of the β-expansion of x and this sequence is denoted by ε(x, β), that is,
ε(x, β) = (ε1(x, β), ε2(x, β), . . . , εn(x, β), . . . ).
Sometimes we write εn(x) instead of εn(x, β) if there is no confusion. In [25], he also
proved that there exists a unique invariant measure µβ which is equivalent to the
Lebesgue measure when β is not an integer (the density formula was given by Gel’fond [12] and Parry [22] independently), while it is well known that the Lebesgue measure is Tβ-invariant when β is an integer. Furthermore, the β-transformation is ergodic and
exponentially mixing with respect to µβ(see Fan etc [9], Philipp [24] and R´enyi [25]).
From the definition of β-digit εn(·, β), we know that the set of possible values of β-digits is A = {0, 1, . . . , β − 1} when β is an integer, otherwise, A = {0, 1, . . . , [β]}.
Let (AN, σ) be the symbolic dynamics with σ the shift transformation on AN. For any word u, v in the symbolic space, uv denotes the concatenation of u and v. Denote ω|n as the prefix of the sequence ω ∈ AN with length n. The finite word un(n ∈ N) and sequence u∞ mean uu| {z }· · · u
n
and uu· · · u · · · respectively. We denote by Dβ the set of the admissible sequences in AN, that is,
Dβ ={ω ∈ AN: there exists some x∈ [0, 1) such that ε(x, β) = ω}.
When β is an integer, Dβ is simply AN (or more precisely AN = Sβ defined below);
when β is not an integer, Dβ was characterized by Parry [22] (see Theorem 2.1 below) by the β-expansion of the number 1, denoted by ε(1, β), which can be obtained in a similar manner as the β-expansion of numbers in [0, 1). We say that ε(1, β) is infinite if there are infinitely many non-zero elements in the sequence ε(1, β), otherwise, it is said to be finite. For finite case, i.e., ε(1, β) =(ε1(1),· · · , εn(1), 0∞
)
with εn(1)̸= 0 for
some n≥ 1, we take ε∗(1, β) = (ε1(1), ε2(1),· · · , εn−1(1), (εn(1)− 1)
)∞
as the infinite expansion of 1. We will still write ε(1, β) instead of ε∗(1, β) for finite cases for the sake of simplicity so that there is no ambiguity in the rest of this paper. To state the following theorem, we give two notations <lex and ≤lex, the lexicographical orders on AN. That is, let ω, ω′ ∈ AN, then ω <
lex ω′ means that there exists n ≥ 1 such that ωn < ωn′ and ωj = ω′j for all j < n, and ω ≤lex ω′ means that ω <lex ω′ or ω = ω′.
Theorem 2.1 ([22]). Let β > 1 be a real number and ε(1, β) the infinite expansion of
the number 1. Then ω ∈ Dβ if and only if
σk(ω) <lex ε(1, β) for all k≥ 0.
Let Sβ be the closure of the set Dβ. It is well known that Sβ = AN when β is an
integer and otherwise, (Sβ, σ|Sβ) is a subshift of (AN, σ), where σ|Sβ is the restriction
Corollary 3 ([4, 16, 22]). Let β > 1 be a real number and ε(1, β) the infinite expansion
of the number 1. Then
Sβ ={ω ∈ AN : σkω≤lex ε(1, β) for all k≥ 0}.
Topological entropy of Tβ and the measure-theoretical entropy of µβ share the
same value log β, and µβ is the unique measure of maximal entropy (see Dajani and
Kraaikamp [7], Hofbauer [13], Ito and Takahashi [14]). In 1989, Blanchard [4] outlined a classification for all numbers β > 1 according to the topological properties of Sβ,
furthermore, the Lebesgue measures and Hausdorff dimensions of all classes were cal-culated by Schmeling [29]. Recently, Li and Wu [17] provided another classification by the quantity ln(β), which is defined as
ln(β) = sup{k ≥ 0 : εn+j(1, β) = 0 for all 1 ≤ j ≤ k} (2.4)
for all n≥ 0. Let
A0 ={β ∈ (1, +∞) : lim sup
n→∞ ln(β) <∞, i.e., {ln(β)} is bounded}, A1 ={β ∈ (1, +∞) : lim sup
n→∞
ln(β)
n = 0 and{ln(β)} is not bounded}, A2 ={β ∈ (1, +∞) : lim sup
n→∞
ln(β) n ̸= 0}.
The key function ln(β) states the maximal length of the string of 0’s following εn in ε(1, β) and that the sets A0, A1, A2 form a disjoint partition of numbers β > 1. All
β’s satisfying Sβ that are subshift of finite type are contained in A0, and moreover,
β ∈ A0 if and only if Sβ satisfies the specification property. Buzzi [6] proved that the
set of β > 1 such that the map Tβ has the specification property is of zero Lebesgue
measure. It is well known that the set A0 has full Hausdorff dimension (see also [29])
and is dense in (1,∞) (see [22]). Denote by Lβ the set of all admissible words, i.e., the language of Dβ. For any admissible word ε1ε2· · · εn, we call
I(ε1, ε2, . . . , εn) ={x ∈ [0, 1) : ε1(x) = ε1, ε2(x) = ε2, . . . , εn(x) = εn},
an n-th cylinder. It is simply to deduce that|I(ε1, ε2, . . . , εn)| ≤ β−nfor any ε1ε2· · · εn∈
Lβ, where|·| denotes the length of an interval. The following proposition characterizes
the sizes of cylinders by the classification in [17].
Proposition 2.1 ([17]). 1. β ∈ A0 if and only if there exists a constant C such that
for all x∈ [0, 1) and n ≥ 1, C 1
2. β ∈ A0∪ A1 if and only if for all x∈ [0, 1),
lim
n→∞−
log|In(x)|
n = log β.
Define a projection function φβ from Sβ to [0, 1] as the following:
φβ(ω) = ∞ ∑ i=1 ωi βi where ω = (ω1, ω2, . . . , ωi, . . . )∈ Sβ. (2.5)
Then φβ is one to one except at the countable many points for which the β-expansions
are finite and the restriction of φβ to which is two to one. It is easy to know that φβ
is continuous and φβ ◦ σ = Tβ ◦ φβ.
The following lemma from [17] describes a way to get full cylinders(the image under
Tβ is the full interval [0, 1)), will be used to prove Lemma 3.1 and Lemma 3.2 below.
Lemma 2.1 ([17]). Let β > 1 be a real number and ε1ε2· · · εn an admissible word.
Denote Mn(β) = max
1≤k≤n{lk(β)}, then for any m > Mn(β), the cylinder
I(ε1, ε2,· · · , εn, 0,| {z }· · · , 0
m
)
is a full interval of rank n + m and its length equals β−(n+m).
2.2
Hausdorff dimensions and measures
Let us recall the definitions of both the Hausdorff measures and dimensions, as well as a useful mass distribution principle which will be used later. A finite or countable collection of subsets{Ui} of R is called a δ-cover of a set E ⊂ R if |Ui| < δ for all i and
E ⊂ ∪∞i=1Ui. Let E be a subset of R and s ≥ 0. For all δ > 0, we define Hs
δ(E) = inf
{ ∞ ∑
i=1
|Ui|s :{Ui} is a δ-cover of E
}
.
The s-dimensional Hausdorff measure of E is defined as
Hs
(E) = lim
δ→0H s δ(E).
We know that there exists a critical point s0 such that Hs(E) = ∞ if s < s0 and
Hs(E) = 0 if s > s
0. This point is called the Hausdorff dimension of E, denoted by
dimHE, that is,
dimHE = inf{s : Hs(E) = 0} = sup{s : Hs(E) =∞}.
The following mass distribution principle is usually used to estimate a lower bound for the Hausdorff dimension of a set. We refer to Falconer [8] and Mattila [18] for further properties of Hausdorff dimension.
Theorem 2.2 (Mass distribution principle). Let E ⊂ R and µ a finite measure with
µ(E) > 0. Suppose that there exist s≥ 0, C > 0 and δ > 0 such that
µ(U )≤ C|U|s (2.6)
for all sets U with |U| ≤ δ, where |U| denote the diameter of the set U. Then
dimHE ≥ s.
Remark 1. In (2.6), we can replace the set U by any ball B(x, r) of radius r centered
at x with r is sufficiently small.
3
Proof of Theorem 1.1
In this section we give a detailed proof for Theorem 1.1. First, we obtain several lemmas for β ∈ A0 and then go on to prove Theorem 1.1 using the approximation
method.
3.1
The case of bases in A
0Let β ∈ A0 and Mβ ≥ max{ln(β) : n≥ 1}. Denote
WN ={0Mβw0Mβ : w ∈ Lβ and |w| = N − 2Mβ},
where 2Mβ ≤ N ∈ N and by WNN the set of sequences u1u2· · · un· · · with un ∈ WN.
Let m∈ N and put
FN,mβ ={x ∈ [0, 1) : ε(x, β) = (0Mβwn0Mβ)
n≥1 ∈ WNN, 0 m
/
∈ wn for all n∈ N}, where 0m ∈ wn/ means that the word 0m does not appear in ωn.
Lemma 3.1. Let β ∈ A0. For any N > 2Mβ and Mβ < m≤ N − 2Mβ, we have
dimH(FN,mβ )≥ sβN,m := N − 2Mβ mN ( log(β− 1) log β − 1 ) + 1− 2Mβ N . (3.7)
Proof. We will show a mass distribution µ supported on FN,mβ and then apply the mass distribution principle.
Firstly, we define the measure µ on the cylinders and then extend it to any Borel set step by step. Let µ([0, 1)) = 1.
Step I. Define µ(I(ε1)) = 1 if ε1 = 0 and otherwise, µ(I(ε1)) = 0.
Step II. Assume µ(I(ε1, . . . , εk−1)) is well defined, we will define µ(I(ε1, . . . , εk−1, εk))
by the following three cases according to the position k. Denote the set
Case (i). k ∈ P . Define µ(I(ε1, . . . , εk−1, εk)) = µ(I(ε1, . . . , εk−1)) if εk = 0 and
otherwise, µ(I(ε1, . . . , εk−1, εk)) = 0.
Case (ii). k = iN + Mβ + j with i = 0, 1, 2, . . . and 1≤ j < m. Let µ(I(ε1, . . . , εk−1, εk)) =
|I(ε1, . . . , εk−1, εk)| |I(ε1, . . . , εk−1)|
µ(I(ε1, . . . , εk−1)). (3.8)
Case(iii). k = iN + Mβ+ j with i = 0, 1, 2, . . . and m ≤ j ≤ N − 2Mβ. If ε1· · · εk−1
does not end up with 0m−1, then µ(I(ε
1, . . . , εk−1, εk)) is defined as the formula (3.8)
and otherwise, µ(I(ε1, . . . , εk−1, εk)) = 0 if εk = 0, |I(ε1,...,εk−1,εk)|
|I(ε1,...,εk−1)\I(ε1,...,εk−1,0)|µ(I(ε1, . . . , εk−1)) if εk ̸= 0. Step III. Continue the procedures in Step II as k → ∞, the measure on cylinders will
be well defined.
Step IV. Extend the measure µ from the cylinders to the Borel field by Kolmogorov
extension theorem.
From the construction of µ, we know that it is supported on FN,mβ .
Secondly, we will prove that the measure µ satisfies the condition (2.6) for any cylinder and any ball.
Step (a). For any 0Mβω0Mβ ∈ WN, we claim that
µ(I(0Mβω0Mβ))≤ ( β β− 1 )N−2Mβ m 1 βN−2Mβ. (3.9)
In fact, by Step (II) Case (i), we know µ(I(0Mβω0Mβ)) = µ(I(0Mβω)). Let j(0 ≤ j ≤
N−2Mβ
m ) be the times that the word 0
m−1 appears in ω and
ω = ε1· · · εi10m−1εi1+m· · · εij0
m−1ε
ij+m· · · εN−2Mβ.
That is, il(1 ≤ l ≤ j) are just the positions which the word 0m−1 follows. Therefore,
written as |I(0Mβε 1· · · εN−2Mβ)| |I(0Mβε 1· · · εN−2Mβ−1)| · · · |I(0Mβε1· · · εij+m+1)| |I(0Mβε 1· · · εij0 m−1ε ij+m)| µ(I(0Mβε 1· · · εij0 m−1 εij+m)) = |I(0 Mβε 1· · · εN−2Mβ)| |I(0Mβε 1· · · εij0 m−1ε ij+m)| |I(0Mβε 1· · · εij0 m−1ε ij+m)| · µ(I(0 Mβε 1· · · εij0 m−1)) |I(0Mβε 1· · · εij0 m−1)|\|I(0Mβε 1· · · εij0 m−10)| = |I(0Mβε 1· · · εN−2Mβ)| |I(0Mβε 1· · · εij0 m−1)| β−(Mβ+ij+m−1)− β−(Mβ+ij+m) µ(I(0Mβε 1· · · εij0 m−2)) |I(0Mβε 1· · · εij0 m−2)| = β β− 1 |I(0Mβε 1· · · εN−2Mβ)| |I(0Mβε 1· · · εij0 m−2)|µ(I(0 Mβε 1· · · εij0 m−2)) = · · · = ( β β− 1 )j |I(0Mβε 1· · · εN−2Mβ)| |I(0Mβ)| µ(I(0 Mβ)) ≤ ( β β− 1 )N−2Mβ m 1 βN−2Mβ,
where the last inequality holds because j ≤ N−2Mβ
m , |I(0 Mβε
1· · · εN−2Mβ)| ≤ β
−(N−Mβ),
|I(0Mβ)| = β−Mβ and µ(I(0Mβ)) = 1. Thus (3.9) holds.
Step (b). We will prove that (2.6) holds for any cylinder I(ε1,· · · , εk). Without loss of
generality, assume ε1· · · εkis the prefix of some x∈ FN,mβ , otherwise, (2.6) will naturally
hold since µ(I(ε1,· · · , εk)) = 0. Note that I(ε1,· · · , εk)⊂ I(0Mβω10Mβ· · · 0Mβω[k N]0 Mβ), we know µ(I(ε1,· · · , εk)) ≤ µ(I(0Mβω10Mβ· · · 0Mβω[Nk]0 Mβ)) = µ(I(0Mβω 10Mβ))· · · µ(I(0Mβω[k N]0 Mβ)) ≤ ( β β− 1 )N−2Mβ m 1 βN−2Mβ [Nk] ≤ ( C 1 βk )SN,mβ ≤ (|I(ε1,· · · , εk)|)S β N,m,
where the equality is from the construction of the measure µ, the second inequality is derived from (3.9) and C is an absolute constant in Proposition 2.1.
Step (c). For any ball B(x, r), there exists n ∈ N such that β−k−1 < r ≤ β−k, then
B(x, r) can be covered by at most four k-th adjoint cylinders. By applying this result
in step (b), we know that (2.6) holds for any ball.
Finally, the application of the mass distribution principle (Theorem 2.2) implies (3.7).
Let
The set Fβ
m can be written as the following type of badly approximable Fmβ ={x ∈ [0, 1) : Tβnx≥ β−m for all n∈ N}.
The general badly approximable set for β = 2 was studied in [19]. Note that FN,mβ ⊂
F2m+2Mβ
β, letting N → ∞ in Lemma 3.1, we obtain the following proposition. Proposition 3.1. Let β ∈ A0. For any m > 2Mβ, we have
dimH(Fmβ)≥ 2 m− 2Mβ ( log(β − 1) log β − 1 ) + 1.
Lemma 3.2. Let β ∈ A0 and J be a closed interval. Then
dimH(E β J ∩ F β 3m)≥ sβm := 1 m ( log(β − 1) log β − 1 ) + 1 for any m≥ 2Mβ.
Proof. Choose a sequence{an} in J such that {an} is dense in J and |an+1−an| ≤ n+11
and let bn = [en(an+n
− 12)
], where [·] represents the integer part of a real number. For any given N ∈ N with N > m − 2Mβ, we can obtain recursively a sequence {cn} of natural numbers such that
bn ≤ N + N n ∑ i=1 ci+ n ∑ i=1 (i + Mβ+ 1) < bn+ N. (3.10)
It is simple to check that such {cn} can be uniquely determined. Denote
dn= N + N n ∑ i=1 ci+ n−1 ∑ i=1 (i + Mβ+ 1).
We will now define the function f on FN,mβ . For any x∈ FN,mβ with its β-expansion
ε(x, β) = (0Mβw
n0Mβ)n≥1, we firstly construct a sequence {ξ∗} from ε(x, β). Write ξ(0) = (ξi(0)) = ε1ε2· · · εN0Mβw10Mβ0Mβw20Mβ· · · 0Mβwn0Mβ· · · ,
where ε1ε2· · · εN is the prefix of ε(1, β) with length N , that is, ξ(0)is obtained by adding
the word ε1ε2· · · εN before ε(x, β), it works according to Lemma 2.1 since ε(x, β) begins
with the string of 0’s with length Mβ. Choose u1 := ξ(0)|10Mβv1 with v1 ̸= ξ (0) 1+Mβ+1.
Let
ξ(1) = (ξi(1)) = ξ(0)|d1u10Mβwc1+10Mβ0Mβwc1+20Mβ · · · ,
that is, insert the word u1 between the positions d1 and d1+ 1 of ξ(0). Assume ξ(k−1)
is well defined, we obtain ξ(k) according to inserting u
k := ξ(k−1)|k0Mβvk with vk ̸= ξk+M(k−1)
β+1 between the positions dk and dk+ 1 of ξ
(k−1), that is,
ξ(k)= (ξi(k)) = ξ(k−1)|dkuk0Mβwck+10
Mβ0Mβw
ck+20
As this procedure continues, we get a sequence {ξ(k)}k
≥1 with ξ(k)|dk = ξ
(k−1)|d
k for all
k ≥ 2 and denote ξ∗ = (ξi∗) as the limit point of the sequence {ξ(k)}. That is,
ξ∗ = ε1· · · εN0Mβω10Mβ· · · 0Mβωc10Mβu10Mβωc1+10Mβ· · · 0Mβωcn0
Mβu
n0Mβωcn+10
Mβ· · · .
According to Theorem 2.1, we know ξ(k) ∈ Dβ and ξ∗ ∈ Sβ. Denote
x∗ = φβ(ξ∗) = ξ∗1 β + ξ∗2 β2 +· · · + ξn∗ βn +· · · . Then ε(x∗, β) = ξ∗. We claim that dn−Mβ ≤ τ β n(x∗)≤ dn for all n > N. (3.11)
Yet in fact, we have τβ
n(x∗)≤ dn since ε(x∗, β) = ξ∗ and σdn(ξ∗) = u
n0Mβwcn+10
Mβ· · · = ξ∗|n0Mβv
n0Mβwcn+10
Mβ· · · from the construction of ξ∗
and x∗. All that remains to be proven is τnβ(x∗) ≥ dn−Mβ, since the word ξ∗|n does not appear in any first dn−Mβ positions of ξ
∗ except the initial position. From the
structure of x and the construction of ξ∗, we know that ξ∗|n does not appear in the positions lying in any 0Mβwi0Mβ(i≥ 1) since ξ∗|n begins with the first N digits of the
β-expansion of the number 1 and the maximal length of the string of 0’s in ξ∗|N is less than Mβ. Combining |ui| = i + 1 + Mβ < n for all 1 ≤ i < n − Mβ − 1 and because
the last letter of ui is not the same with ξi+M∗ β+1, we know that ξ
∗|n does not appear
in ui(1≤ i < n − Mβ). So (3.11) holds. We claim that lim n→∞ ( log τβ n(x∗) n − an ) = 0, (3.12) which implies A ( log τβ n(x∗) n ) = J (3.13)
since {an} is dense in J. Indeed, by (3.10), we have
bn ≤ dn+ (n + Mβ+ 1) < bn+ N. (3.14)
Combined (3.14) and (3.11), to obtain
bn−Mβ − (n + 1) ≤ τ
β
n(x∗)≤ bn+ N− (n + Mβ+ 1) ≤ bn (3.15)
whenever n > N . Note that because bn= [en(an+n
− 12) ], we know lim n→∞ ( log bn n − an ) = 0 and lim n→∞ ( log(bn−Mβ − (n + 1)) n − an ) = lim n→∞(an−Mβ − an).
By|an+1−an| ≤ n+11 , we have obtained limn→∞(an−Mβ−an) = 0. Therefore, by (3.15),
Define the function f as f (x) = x∗ for any x ∈ FN,mβ . Combining (3.13) and the structure of ξ∗, we know
f (FN,mβ )⊂ EJβ∩ F3mβ (3.16) whenever m ≥ 2Mβ.
We consider f−1 on f (FN,mβ ) as f−1(x∗) = x, that is, delete the first N digits and the digits between di+ 1 and di+ i + 1 + Mβ positions for all i≥ 1. To put this more
clearly, the words ε1· · · εN and un(n ≥ 1) are removed from ξ∗. We claim that f−1 is
(1−ε)-H¨older for any ε > 0. In fact, for any x∗, y∗ ∈ In(x∗), then x, y ∈ In′(x) for some
n′ from the definition of f−1. By (3.10), we know dnis of exponential rate and that the
deleted digits |un| have a of polynomial rate, therefor we can be assured n′ ≥ n(1 − ε) when n is large enough. Since |x∗− y∗| ≤ β−n and |x − y| ≤ β−n′ ≤ β−n(1−ε), so
|f−1(x∗)− f−1(y∗)| ≤ |x∗− y∗|1−ε . Therefore, dimH(F β N,m)≤ 1 1−εdimHf (F β
N,m). Let ε→ 0, by (3.16) and Lemma 3.1, we
have
dimH(EJβ∩ F3mβ )≥ s
β N,m.
Let N → ∞, we obtain dimH(EJβ ∩ F3mβ )≥ sβm.
Remark 2. Since EJβ ∩ F3mβ ⊂ EJβ, Lemma 3.2 implies that Theorem 1.1 holds when β ∈ A0 by letting m→ ∞.
3.2
The general case for any base
Let 1 < β′ ≤ β. Since Dβ′ ⊂ Dβ, we know that φβ(Dβ′) is a Cantor-like subset of [0, 1).
Define a function h : φβ(Dβ′)→ [0, 1) as h(x) = ε1(x, β) β′ + ε2(x, β) β′2 +· · · + εn(x, β) β′n +· · · ,
that is, h(x) = φβ′(ε(x, β)). The function is well defined since ε(x, β)∈ Dβ′.
Lemma 3.3. For any x∈ φβ(Dβ′), we have ε(x, β) = ε(h(x), β′).
Proof. Since ε(x, β) ∈ Dβ′, we have ε(φβ′(ε(x, β)), β′) = ε(x, β) by the uniqueness of
β-expansion. Thus ε(h(x), β′) = ε(x, β) based on the definition of h.
Lemma 3.4. The function h is bijective and strictly increasing.
Proof. Suppose h(x) = h(y), then ε(x, β) = ε(h(x), β′) = ε(h(y), β′) = ε(y, β) by Lemma 3.3, which implies x = y. That is, h is injective. For any z ∈ [0, 1), take
x = φβ(ε(z, β′)). It is simple to check h(x) = z, that is, h is surjective.
Suppose x < y, then ε(x, β) <lex ε(y, β). Thus h(x)≤ h(y) since φβ′ is increasing in Dβ′ and ε(x, β), ε(y, β) ∈ Dβ′. Therefore h(x) < h(y) based on the injection of h.
Remark 3. Since ε(·, β) is right continuous and φβ′(·) is continuous, we know that h
is right continuous.
Lemma 3.5. The function h is H¨older continuous on Fmβ, moreover,
|h(x) − h(y)| ≤ (β + 1)βm|x − y|log β′log β (3.17) for any x, y ∈ Fβ
m∩ φβ(Dβ′).
Proof. Without loss of generality, we assume x > y since it is similar for the case x < y and (3.17) is trivial if x = y. Let n ≥ 1 be the smallest integer such that εn(x, β) > εn(y, β). Thus |x − y| = εn(x, β)− εn(y, β) βn + 1 βn( εn+1(x, β) β +· · · ) − 1 βn( εn+1(y, β) β +· · · ) ≥ 1 βn + 1 βn+m − 1 βn = 1 βn+m,
where the inequality comes from the facts that εn(x, β)− εn(y, β) ≥ 1, the sequence εn+1(x, β)εn+2(x, β)· · · does not contain 0m, and that
εn+1(y,β)
β +
εn+2(y,β)
β2 +· · · ≤ 1. By
Lemma 3.4, we know h(x) > h(y). Application of Lemma 3.3 implies
|h(x) − h(y)| = εn(x, β)− εn(y, β) β′n + 1 β′n( εn+1(x, β) β′ +· · · ) − 1 β′n( εn+1(y, β) β′ +· · · ) ≤ β β′n + 1 β′n = β + 1 β′n ,
where the inequality holds because εn(x, β)− εn(y, β)≤ β,
εn+1(x,β) β′ + εn+2(x,β) β′2 +· · · ≤ 1 and εn+1(y,β) β′ + εn+2(y,β)
β′2 +· · · ≥ 0. Therefore, we obtain (3.17) from the estimations of |x − y| and |h(x) − h(y)|. Denote EJβ,β′ ={x ∈ φβ(Dβ′) : A ( log τnβ(x) n ) = J}.
Lemma 3.6. For any given closed interval J and m∈ N, we obtain
h(EJβ,β′∩ Fmβ) = EJβ′ ∩ Fmβ′.
Proof. According to Lemma 3.3, we know τnβ′(h(x)) = τnβ(x) for any x∈ φβ(Dβ′) and
n ∈ N. Thus h(EJβ,β′)⊂ EJβ′. Note that since h(Fmβ ∩ φβ(Dβ′))⊂ Fmβ′, we can obtain
h(EJβ,β′∩ Fmβ)⊂ EJβ′ ∩ Fmβ′.
Meanwhile, for any y ∈ EJβ′∩ Fmβ′, note that h is bijective, take z = h−1(y)∈ φβ(Dβ′).
We obtain ε(z, β) = ε(h(z), β′) = ε(y, β′) by Lemma 3.3, by z ∈ EJβ,β′ ∩ Fmβ, which implies
Finally, we will summarize the proof of Theorem 1.1.
Proof of Theorem 1.1. Let β′ ∈ A0 and β′ ≤ β. According to Lemma 3.5 and Lemma
3.6, note that EJβ,β′ ⊂ EJβ, we have dimH(E β′ J ∩F β′ 3m) = dimH(h(E β,β′ J ∩F β 3m))≤ log β log β′ dimH(E β,β′ J ∩F β 3m)≤ log β log β′ dimH(E β J∩F β 3m).
That is, dimH(EJβ ∩ F β 3m)≥ log β′ log β dimH(E β′ J ∩ F β′ 3m). By applying Lemma 3.2 to β′, we obtain dimH(EJβ∩ F β 3m)≥ log β′ log βs β′ m. Thus dimH(E β J)≥ log β′ log βs β′ m
since EJβ ⊃ EJβ∩ F3mβ . By letting m→ ∞, we obtain dimHEJβ ≥
log β′ log β.
Since A0 is dense in (1,∞), let β′ → β, and we obtain dimHEJβ = 1.
Acknowledgement. The first author is partially supported by the National
Sci-ence Council, ROC (Contract No NSC 99-2115-M-259-003) and National Center for Theoretical Sciences. The second author is partially supported by ”Fundamental Re-search Funds for the Central Universities” (Contract No 2011ZM0083). This work was partially carried out when the second author visited CMTP, National Central Univer-sity. He would like to thank the institute for their warm hospitality.
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J.-C. Ban: Department of Mathematics, National Dong Hwa University, Hualien 970003, Taiwan
E-mail: jcban@mail.ndhu.edu.tw
B. Li: Department of Mathematics, South China University of Technology, Guangzhou 510640, P. R. China
