On the bifurcation set of unique expansions

# On the bifurcation set of unique expansions

## Abstract.

Given a positive integer , for let be the set of having a unique -expansion with the digit set , and let be the set of corresponding -expansions. Recently, Komornik et al. showed in [21] that the topological entropy function is a Devil’s staircase in .

Let be the bifurcation set of defined by

 B={q∈(1,M+1]:H(p)≠H(q)for anyp≠q}.

In this paper we analyze the fractal properties of , and show that for any ,

 limδ→0dimH(B∩(q−δ,q+δ))=dimH\pazocalUq,

where denotes the Hausdorff dimension. Moreover, when the univoque set is dimensionally homogeneous, i.e., for any open set that intersect .

As an application we obtain a dimensional spectrum result for the set containing all bases such that admits a unique -expansion. In particular, we prove that for any we have

 dimH(U∩(1,t])=maxq≤tdimH\pazocalUq.

We also consider the variations of the sets when changes.

###### Key words and phrases:
Bifurcation set; Univoque sets; Univoque bases; Hausdorff dimensions; Devil’s staircase.
###### 2010 Mathematics Subject Classification:
Primary:37B10, Secondary: 11A63, 28A78

## 1. Introduction

Fix a positive integer . For any each has a -expansion, i.e., there exists a sequence with each such that

 (1.1) x=∞∑i=1xiqi=:πq((xi)).

The sequence is called a -expansion of . If no confusion arises the alphabet is always assumed to be .

Non-integer base expansions have received a lot of attention since the pioneering papers of Rényi [32] and Parry [31]. It is well known that for any Lebesgue almost every has a continuum of -expansions (cf. [33, 9]). Moreover, for any there exist and such that has precisely different -expansions (see e.g., [17, 35]). For more information on non-integer base expansions we refer the reader to the survey paper [20] and the references therein.

In this paper we focus on studying unique -expansions. For let

 \pazocalUq:={x∈Iq,M:x has a unique q-% expansion},

and let be the set of corresponding -expansions. These sets have been the object of study in many articles and have a very rich topological structure (see for example [23, 13]). Komornik et al. studied in [21] the Hausdorff dimension of , and showed that the dimension function has a Devil’s staircase behavior. Moreover, they showed that the entropy function

 H:(1,M+1]→[0,log(M+1)];q↦htop(Uq)

is a Devil’s staircase (see Lemma 2.4 below). Recently, Alcaraz Barrera et al. investigated in [1] the dynamical properties of , and determined the maximal intervals on which the entropy function is constant.

Let be the bifurcation set of the function defined by

 B={q∈(1,M+1]:H(p)≠H(q) for any p≠q}.

Then is the set of bases where the entropy function is not locally constant. In [1] Alcaraz Barrera et al. gave a characterization of and showed that has full Hausdorff dimension. In particular, we have

 (1.2) B=(qKL,M+1]∖⋃[pL,pR],

where is the Komornik-Loreti constant (cf. [22]) and the union on the right hand side is countable and pairwise disjoint (see Section 2 below for more explanation).

From [13] we know that the univoque set has a fractal structure and might have isolated points. Our first result states that for the univoque set is dimensionally homogeneous, i.e., the local Hausdorff dimension of equals the full dimension of .

###### Theorem 1.

Let . Then for any open set with we have

 dimH(\pazocalUq∩V)=dimH\pazocalUq.
###### Remark 1.1.

1. Note by (1.2) that . So Theorem 1 implies that the univoque set is dimensionally homogeneous for any .

2. Theorem 1 holds for a even larger set of ’s (see the final remark at end of Section 3).

Throughout the paper we will use to denote the topological closure of a set . Our second result presents a close relationship between the bifurcation set and the univoque sets .

###### Theorem 2.

For any we have

 limδ→0dimH(¯¯¯¯B∩(q−δ,q+δ))=dimH\pazocalUq.
###### Remark 1.2.

1. Since by (1.2) and (2.5) that the difference between and is countable, Theorem 2 also holds if we replace by .

2. Note that for any (see Lemma 2.4 below). So, Theorem 2 gives the following characterization of :

 q∈¯¯¯¯B∖{qKL}⟺limδ→0dimH(¯¯¯¯B∩(q−δ,q+δ))=dimH\pazocalUq>0.

It is well-known that the univoque set has a close connection with the set of univoque bases for which has a unique -expansion with alphabet . For example, in [13] De Vries and Komornik showed that is closed if and only if . The set has many interesting properties itself. Erdős et al. showed in [16] that is an uncountable set of zero Lebesgue measure. Daróczy and Kátai proved in [12] that the Hausdorff dimension of is 1 (see also [21]). Komornik and Loreti showed in [22] that the smallest element of is . In [23] the same authors studied the topological properties of , and showed that its closure is a Cantor set. Recently, Kong et al. proved in [26] that for any we have

 (1.3) dimH(¯¯¯¯U∩(q−δ,q+δ))>0for any% δ>0.

On a different note, in [7] Bonanno et al. introduced a set

 (1.4) Λ={x∈[0,1]:Skx≤x for all n≥0},

where is the tent map defined by and showed that there is a one to one correspondence between the set and the set , where is the set of all rationals with odd denominator. This link is based on work by Allouche and Cosnard (see [2, 4, 5]), who related the set to kneading sequences of unimodal maps. The authors of [7] also explored a relationship between these sets and the real slice of the boundary of the Mandelbrot set.

By using Theorem 2 we investigate the dimensional spectrum of . Our next result strengthens the relationship between and .

###### Theorem 3.

For any we have

 dimH(U∩(1,t])=maxq≤tdimH\pazocalUq.

Moreover, the function is a Devil’s staircase on .

###### Remark 1.3.

1. In [23] it was shown that is a countable set. Hence, Theorem 3 still holds if we replace by .

2. Results from [21] (see Lemma 2.4 below) give that if and only if . In view of Theorem 3 we obtain that for any . This implies that the Hausdorff dimension of is concentrated on the neighborhood of .

As an application of Theorem 3 we investigate the variations of when the parameter changes. For , let be the set of bases such that has a unique -expansion with respect to the alphabet . Theorem 4 characterizes the Hausdorff dimensions of the intersection and the difference . Indeed, we prove the following stronger result.

###### Theorem 4.

1. Let . Then

 dimH(M⋂J=KU(J))=maxq≤K+1dimH\pazocalUq.
2. For any positive integer we have

 dimH(U(L)∖⋃J≠LU(J))=1.
###### Remark 1.4.

By the proof of Theorem 4 it follows that for the intersection

 M⋂J=KU(J)=U(M)∩(1,K+1]

is a proper subset of . This, together with (1.3), implies that for neither the intersection nor the difference set contains isolated points.

We emphasize that for each the univoque set is related to the dynamical system

 Tq,j:[0,Mq−1]→[0,Mq−1];x↦qx−j

for . On the other hand, the set contains all parameters such that has a unique -expansion, and thus is related to infinitely many dynamical systems. A similar set up involving a bifurcation set for infinitely many dynamical systems is considered in [7] (see also [8]). They considered the bifurcation set of an entropy map for a family of maps , called -continued fraction transformations [30], where for each the map is defined by

 (1.5) Tα(x)={1|x|−⌊1|x|+1−α⌋ifx≠0;0ifx=0.

Each map has a unique invariant measure that is absolutely continuous with respect to the Lebesgue measure. They showed that the map

 ψ:α↦hμα(Tα),

assigning to each the measure theoretic entropy , has countably many intervals on which it is monotonic. The complement of the union of these intervals in , i.e., the bifurcation set of denoted by , has Lebesgue measure 0 (see [27] and [8]) and Hausdorff dimension 1 (see [7]). Moreover, in [7] the authors identified a homeomorphism between the set and the set from (1.4), giving also a relation to the set . In [7], however, no information is given on the local structure of . Recently, Dajani and the first author identified in [10] another set that is linked to the sets , and . They investigated a family of symmetric doubling maps , given by

 Sγ(x)=2x−γ⌊2x⌋,

where denotes the integer part of , and showd that the set of parameters for which the map does not have a piecewise smooth invariant density is homeomorphic to . Therefore, the results obtained in this paper about the set can be used to investigate the bifurcation sets , and the set .

The rest of the paper is arranged in the following way. In Section 2 we fix the notation and recall some properties of unique -expansions. Moreover, we recall from [1] some important properties of the bifurcation set . In Section 3 we give the proof of Theorem 1 for the dimensional homogeneousness of . In Section 4 we prove an auxiliary proposition that will be used to prove Theorem 2 in Section 5. The proof of Theorems 3 and 4 will be given in Sections 6 and 7, respectively. We end the paper with some remarks.

## 2. Unique expansions and bifurcation set

In this section we recall some properties of unique -expansions and of the bifurcation set as well. First we need some terminology from symbolic dynamics (cf. [28]).

### 2.1. Symbolic dynamics

Given a positive integer , let denote the set of all finite strings of symbols from , called words, together with the empty word denoted by . Let be the set of sequences with each element . Let be the left shift on defined by . Then is a full shift. For a word we denote by the -fold concatenation of to itself and by the periodic sequence with period block . Moreover, for a word with we denote by the word

 c+=c1…cn−1(cn+1).

Similarly, for a word with we write . For a sequence we denote its reflection by

 ¯¯¯¯¯¯¯¯¯(di)=(M−d1)(M−d2)⋯.

Accordingly, for a word we denote its reflection by .

On words and sequences we consider the lexicographical ordering or is defined as follows. For two sequences we say that if there exists such that and . Moreover, we write if or . Similarly, we write if , and if . We extend this definition to words in the following way. For two words we write if . Accordingly, for a sequence and a word we say if .

Let and let be the set of those sequences that do not contain any word from . We call the pair a subshift. If can be chosen to be a finite set, then is called a subshift of finite type. For we denote by the set of words of length occurring in sequences of . In particular, for we set . The languange of is then defined by

 \pazocalL(X)=∞⋃n=0\pazocalLn(X).

So, is the set of all finite words occurring in sequences from .

For a subshift and a word let be the follower set of in defined by

 (2.1) FX(ω):={(di)∈X:d1…d|ω|=ω},

where denotes the length of a word .

A subshift is called topologically transitive (or simply transitive) if for any two words there exists a word such that . In other words, in a transitive subshift any two words can be “connected” in .

The topological entropy of a subshift is a quantity that indicates its complexity. It is defined by

 (2.2) htop(X)=limn→∞log#\pazocalLn(X)n=infn≥1log#\pazocalLn(X)n,

where denotes the cardinality of a set . Accordingly, we define the topological entropy of a follower set by changing to in (2.2) if the corresponding limit exists. Clearly, if is a transitive subshift, then for any .

### 2.2. Unique expansions

In this subsection we recall some results about unique expansions. For more information on this topic we refer the reader to the survey papers [34, 20] or the book chapter [14]. For , let

 α(q)=α1(q)α2(q)…

be the quasi-greedy -expansion of (cf. [11]), i.e., the lexicographically largest -expansion of not ending with a string of zeros. The following characterization of quasi-greedy expansions was given in [6, Theorem 2.2].

###### Lemma 2.1.

The map is a strictly increasing bijection from onto the set of all sequences not ending with and satisfying

 an+1an+2…⪯a1a2…wheneveran

Recall from (1.1) the definition of the projection map for mapping onto the interval . In general, is not bijective. However, is a bijection between and . The following lexicographical characterization of , or equivalently , was essentially due to Parry [31] (see also [6]).

###### Lemma 2.2.

Let . Then if and only if

 xn+1xn+2…≺α(q) wheneverxn0.

Observe that . As a corollary of Lemma 2.2 we have the following characterizations of and .

###### Lemma 2.3.

1. if and only if the quasi-greedy expansion satisfies

 ¯¯¯¯¯¯¯¯¯¯α(q)≺σn(α(q))≺α(q)for % any n≥1.
2. if and only if the quasi-greedy expansion satisfies

 ¯¯¯¯¯¯¯¯¯¯α(q)≺σn(α(q))≼α(q)for any n≥1.
###### Proof.

Part (i) was shown in [15, Theorem 2.5] and Part (ii) was proven in [15, Theorem 3.9]. ∎

In [13] it was shown that is not necessarily a subshift. Inspired by [21] we consider the set which contains all sequences satisfying

 ¯¯¯¯¯¯¯¯¯¯α(q)≼σn((xi))≼α(q)for alln≥0.

Then is a subshift (cf. [21, Lemma 2.6]). Furthermore, Lemma 2.1 implies that the set-valued map is increasing, i.e., whenever .

Recall that the Komornik-Loreti constant is the smallest element of , which is defined in terms of the classical Thue-Morse sequence . Here the sequence is defined as follows (cf. [3]): , and if has already been defined for some , then . Then the Komornik-Loreti constant is the unique base satisfying

 (2.3) α(qKL)=λ1λ2…,

where

 λi={k+τi−τi−1ifM=2k,k+τiifM=2k+1,

for each . We emphasize that the sequence depends on . By the definition of the Thue-Morse sequence it follows that (cf. [1])

 (2.4) λ2n+1…λ2n+1=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯λ1…λ2n+for anyn≥0.

Recall that a function is called a Devil’s staircase (or Cantor function) if is a continuous and increasing function with , and is locally constant almost everywhere. The next lemma summarizes some results from [21] on the Hausdorff dimension of .

###### Lemma 2.4.

1. For any we have

 dimH\pazocalUq=htop(Vq)logq.
2. The entropy function is a Devil’s staircase in :

• is increasing and continuous in ;

• is locally constant almost everywhere in ;

• if and only if . Moreover, if and only if .

###### Remark 2.5.

1. Lemma 2.4 implies that the dimensional function has a Devil’s staircase behavior: (i) is continous in ; (ii) almost everywhere in ; (iii) for any and for .

2. In [21, Lemma 2.11] the authors showed that is locally constant on the complement of , i.e., for any .

### 2.3. Bifurcation set

In this subsection we recall some recent results obtained in [1], where the authors investigated the maximal intervals on which is locally constant, called entropy plateaus (or simply called plateaus). For convenience of the reader we adopt much of the notation from [1]. We hope that this helps the interested reader who wants to access the relevant background information. Let be the complement of these plateaus. From Lemma 2.4 (ii) we have

 B={q∈(1,M+1]:H(p)≠H(q)for any p≠q}.

Note by (1.2) that is not closed. For the closure we have

 ¯¯¯¯B={q∈(1,M+1]:∀δ>0,∃p∈(q−δ,q+δ) such that H(p)≠H(q)}.

In [1] was denoted by . The following lemma for , the first part of which follows from Remark 2.5 (2), was established in [1, Theorem 3].

###### Lemma 2.6.

, and is a Cantor set of full Hausdorff dimension.

By Lemma 2.4 it follows that and . Since is a Cantor set, we can write

 (2.5) (qKL,M+1]∖¯¯¯¯B=⋃(pL,pR),

where the union is pairwise disjoint and countable. By the definition of it follows that the intervals are the plateaus of . In particular, since is increasing, these intervals have the property that if and only if . This implies that the bifurcation set can be rewritten as in (1.2), i.e.,

 B=(qKL,M+1]∖⋃[pL,pR].

By (2.5) and (1.2) it follows that is countable. The fact that does not have isolated points gives the following lemma (see also [1]).

###### Lemma 2.7.

1. For any there exists a sequence of plateaus such that as .

2. For any there exists a sequence of plateaus such that as .

So, by (2.5), (1.2) and Lemma 2.7 it follows that is a countable and dense subset of . In particular, the set of left endpoints of all plateaus of is dense in .

In [1] more detailed information on the structure of the plateaus of is given. Before we can give the necessary details, we have to recall some notation from [1]. Let be the set of sequences satisfying the inequalities

 ¯¯¯¯¯¯¯¯¯(ai)≼σn((ai))≼(ai)for alln≥0.

In [1, Lemma 3.3] it is proved that the subshift is not transitive for any , where is the unique base such that

 (2.6) α(qT)={(k+1)k∞ifM=2k,(k+1)((k+1)k)∞ifM=2k+1.

The plateaus of are characterized separately for the cases (A) and (B)

(A). First we recall from [1] the following definition.

###### Definition 2.8.

A sequence is called irreducible if

 a1…aj(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯a1…aj+)∞≺(ai)whenever(a1…a−j)∞∈V.
###### Lemma 2.9.

Let be a plateau of .

1. There exists a word such that

 α(pL)=(a1…am)∞ is irreducible,andα(pR)=a1…a+m(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯a1…am)∞.
2. is a transitive subshift of finite type.

3. There exists a periodic sequence such that for any word we can find a large integer and a word satisfying

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯α1(pL)…αN(pL)≺σj(ηων∞)≺α1(pL)…αN(pL)for % anyj≥0.
###### Proof.

Part (i) follows by [1, Proposition 5.2], and Part (ii) follows by [1, Lemma 5.1 (1)].

For (iii) we take

 ν={kifM=2k,(k+1)kifM=2k+1.

Since , by Lemma 2.1 we have . Then (2.6) gives that

 (2.7) ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯α1(pL)α2(pL)≼¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯α1(qT)α2(qT)≺σj(ν∞)≺α1(qT)α2(qT)≼α1(pL)α2(pL)

for all . Note by (i) that is irreducible. By the proof of [1, Proposition 3.17] it follows that for any word there exist a large integer and a word satisfying

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯α1(pL)…αN(pL)≺σj(ηων∞)≺α1(pL)…αN(pL)for any% 0≤j<|η|+|ω|.

This together with (2.7) proves (iii). ∎

(B). Now we consider plateaus of in . Let be the quasi-greedy -expansion of as given in (2.3). Note that depends on . For let

 (2.8) ξ(n)={λ1…λ2n−1(¯λ1…λ2n−1+)∞ifM=2k,λ1…λ2n(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯λ1…λ2n+)∞ifM=2k+1.

Then , and is strictly decreasing to as . Moreover, [1, Lemma 4.2] gives that for all . We recall from [1] the following definition.

###### Definition 2.10.

A sequence is said to be -irreducible if there exists such that , and

 a1…aj(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯a1…aj+)∞≺(ai)

whenever

###### Lemma 2.11.

Let be a plateau of .

1. There exists a word such that

 α(pL)=(a1…am)∞ is ∗-irreducible,andα(pR)=a1…a+m(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯a1…am)∞.
2. is a subshift of finite type, and it contains a unique transitive subshift of finite type satisfying

3. There exists a periodic sequence such that for any word we can find a large integer and a word satisfying

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯α1(pL)…αN(pL)≺σj(ηων∞)≺α1(pL)…αN(pL)for any% j≥0.
###### Proof.

Part (i) follows from [1, Proposition 5.11], and Part (ii) follows from [1, Lemma 5.9]. Then it remains to prove (iii).

By (i) we know that is a -irreducible sequence. Then there exists such that . Note by (i) and (2.8) that is purely periodic while is eventually periodic. Then . Let

 ν={λ1…λ−2nif% M=2k,λ1…λ−2n+1ifM=2k+1.

Then by the proof of [1, Lemma 5.9] we have . Observe by (2.4) and (2.8) that . Then by using it follows that there exists a large integer such that

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯α1(pL)…αN(pL)≺σj(ν∞)≺α1(pL)…αN(pL)for anyj≥0.

The remaining part of (iii) follows by the proof of [1, Lemma 5.8]. ∎

Finally, the following characterization of was established in [1, Theorem 3].

## 3. Dimensional homogeneity of \pazocalUq

In this section we will prove Theorem