Two bifurcation sets arising from the beta transformation with a hole at 0

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Abstract

Given β(1,2], the β-transformation Tβ:xβx(mod1) on the circle [0,1) with a hole [0,t) was investigated by Kalle et al. (2019). They described the set-valued bifurcation set Eβ{t[0,1):Kβ(t)Kβ(t)t>t},where Kβ(t){x[0,1):Tβn(x)tn0} is the survivor set. In this paper we investigate the dimension bifurcation set β{t[0,1):dimHKβ(t)dimHKβ(t)t>t},where dimH denotes the Hausdorff dimension. We show that if β(1,2] is a multinacci number then the two bifurcation sets β and Eβ coincide. Moreover we give a complete characterization of these two sets. As a corollary of our main result we prove that for β a multinacci number we have dimH(Eβ[t,1])=dimHKβ(t) for any t[0,1). This confirms a conjecture of Kalle et al. for β a multinacci number.

Introduction

Given β(1,2], the β-transformation Tβ on the circle RZ[0,1) is defined by Tβ:[0,1)[0,1);xβx(mod1).Following the pioneering work of Rényi [10] and Parry [8] there has been a great interest in the study of Tβ. In general, the system Φβ=([0,1),Tβ) does not admit a Markov partition (cf. [11]), this makes describing the dynamics of Φβ more challenging.

When β=2, Urbański considered in [13], [14] the open dynamical system under the doubling map T2 with a hole at zero. More precisely, for t[0,1) let K2(t)x[0,1):T2n(x)tn0.Here we use a slightly different definition of K2(t) from that by Urbański. By [13, Theorem 1 and Corollary 1] it follows that the dimension function tη2(t)dimHK2(t) is a Devil’s staircase on [0,1), that is (i) η2 is decreasing and continuous on [0,1); (ii) η2 is locally constant almost everywhere on [0,1); and (iii) η2 is not constant on [0,1). Here and throughout the paper dimH denotes the Hausdorff dimension. Moreover, Urbański investigated the bifurcation sets E2t[0,1):K2(t)K2(t)t>tand2t[0,1):η2(t)η2(t)t>t. Clearly, 2E2. It can be easily deduced from the proof of Theorem 1 in [13] that 2=E2, and its topological closure 2¯ is a Cantor set, i.e., a non-empty compact set that has neither isolated nor interior points. Furthermore, the following local dimension property was shown to hold: limr0dimH(E2(tr,t+r))=η2(t) for all tE2. Recently, Carminati and Tiozzo in [1] showed that the local Hölder exponent of the dimension function η2 at any tE2 equals η2(t).

Inspired by the work of Urbański [13], [14], Kalle et al. in [5] considered the analogous problem for the β-transformation with a hole [0,t). More precisely, for t[0,1) they investigated the survivor set Kβ(t)x[0,1):Tβn(x)tn0,and showed that the dimension function tdimHKβ(t) is also a Devil’s staircase on [0,1). Furthermore, they characterized the set-valued bifurcation set Eβt[0,1):Kβ(t)Kβ(t)t>t,and proved that Eβ is a Lebesgue null set of full Hausdorff dimension for any β(1,2). Note that the bifurcation set Eβ defined here coincides with Eβ+{t[0,1):Tβn(t)tn0} in [5] (see also Proposition 2.3(i)). Interestingly, they showed that Eβ contains infinitely many isolated points for Lebesgue almost every β(1,2). This is in contrast to the case where β=2 and E2 has no isolated points. For β-transformation with an arbitrary hole we refer to the work of Clark [2]. We also mention that the study of bifurcation sets plays an important role in one-dimensional dynamics (cf. [4]).

Since for each β(1,2) the dimension function ηβ:tdimHKβ(t) is a Devil’s staircase, it is natural to consider the dimension bifurcation set βt[0,1):ηβ(t)ηβ(t)t>t.This set records those t for which the dimension function ηβ has a ‘change’ within any right neighborhood. Since ηβ is continuous, β cannot have isolated points. On the other hand, the set-valued bifurcation set Eβ contains (infinitely many) isolated points for Lebesgue almost every β(1,2). So in general we cannot expect the coincidence of the two bifurcation sets β and Eβ. That being said, in this paper we show that if β is a multinacci number, i.e., the unique root in (1,2) of the equation xm+1=xm+xm1++x+1for some mN, then the two bifurcation sets indeed coincide. Importantly, if β is a multinacci number then its quasi-greedy expansion of 1 is of the form ((1m0)). This property will be useful in our analysis. Here for β(1,2] the quasi-greedy β-expansion δ(β)=δ1(β)δ2(β) of 1 is the lexicographically largest zero–one sequence not ending with an infinite string of zeros and satisfying 1=i=1δi(β)βi (see Section 2 for more details). Furthermore, throughout the paper we will use lexicographical order ‘,,’ and ‘’ between sequences and words.

When β(1,2) is a multinacci number, the following result for the set-valued bifurcation set Eβ was established in [5, Theorems C and D]. We record it here for later use.

Theorem 1.1 [5]

Let β(1,2] be a multinacci number. Then the topological closure Eβ¯ is a Cantor set. Furthermore, maxEβ¯=11β.

In order to give a complete description of the dimension bifurcation set β we introduce a class of basic intervals.

Definition 1.2

Let β(1,2]. A word s1sm is called β-Lyndon if si+1sms1smi1i<m,andσn((s1sm))δ(β)n0.Accordingly, an interval [tL,tR)[0,1) is called a β-Lyndon interval if there exists a β-Lyndon word s1sm such that tL=i=1msiβiandtR=βmβm1tL.

Here we mention that in Definition 1.2 the left endpoint tL=(s1sm0)β has a finite β-expansion and the right endpoint tR=((s1sm))β has a periodic β-expansion, see Section 2 for more explanations.

We will show that the β-Lyndon intervals are pairwise disjoint for all β(1,2], and when β is multinacci they cover the interval [0,11β) up to a Lebesgue null set. The latter statement can be seen as a consequence of our main result for the coincidence of the two bifurcation sets, which we state below.

Theorem 1

Let β(1,2] be a multinacci number. Then β=Eβ=0,11β[tL,tR)=t[0,1):limr0dimH(β(t,t+r))=dimHKβ(t)>0, where the union is taken over all pairwise disjoint β-Lyndon intervals.

By Theorem 1 it follows that the topological closure [tL,tR] of each β-Lyndon interval is indeed a maximal interval where the dimension function ηβ is constant. As a corollary of Theorem 1 we confirm a conjecture of [5] for β a multinacci number.

Corollary 2

If β(1,2] is a multinacci number, then dimH(Eβ[t,1])=dimHKβ(t)t[0,1).

The rest of the paper is organized as follows. In Section 2 we recall some properties from symbolic dynamics and the dimension formula for the survivor set Kβ(t). The proof of Theorem 1 and Corollary 2 will be given in Section 3. In Section 4 we make some remarks and point out that the method of proof for Theorem 1 can be applied to some other special values of β(1,2].

Section snippets

Preliminaries and β-Lyndon intervals

Given β(1,2], for each xIβ[0,1(β1)] there exists a sequence (di)=d1d20,1N such that x=i=1diβi((di))β.The sequence (di) is called a β-expansion of x. Sidorov [12] showed that for β(1,2) Lebesgue almost every xIβ has a continuum of β-expansions. This is rather different from the case when β=2 where every number in I2=[0,1] has a unique dyadic expansion except for countably many points that have precisely two expansions. Given xIβ, among all of its β-expansions let b(x,β)=(bi(x,β))be

Proof of Theorem 1

In this section we will prove Theorem 1. First we show that the dimension bifurcation set β coincides with the set-valued bifurcation set Eβ, we then derive a complete characterization of these sets via the β-Lyndon intervals. The proof heavily relies upon the transitivity of the symbolic survivor set K˜β(t) (see Lemma 3.2).

Proposition 3.1

Let β(1,2) be a multinacci number. Then β=Eβ=0,11β[tL,tR),where the union is taken over all β-Lyndon intervals.

Observe by Lemma 2.2 that the β-Lyndon intervals are

Final remarks

The main results obtained in this paper can be easily modified to study the following analogous bifurcation sets: Eβt[0,1):Kβ(t)Kβ(t)tt,βt[0,1):dimHKβ(t)dimHKβ(t)tt. If β(1,2] is a multinacci number, one can show that β=Eβ=0,11β[tL,tR]=t[0,1):limr0dimH(Eβ(tr,t))=limr0dimH(Eβ(t,t+r))=dimHKβ(t)>0, where the union is taken over all pairwise disjoint closed β-Lyndon intervals.

Observe that the main result Theorem 1 holds under the assumption that β(1,2] is a multinacci

Acknowledgments

The authors thank the anonymous referee for many useful remarks and suggestions. Both authors were supported by an LMS Scheme 4 grant. The first author was supported by EPSRC, UK grant EP/M001903/1. The second author was supported by NSFC, PR China No. 11971079 and the Fundamental and Frontier Research Project of Chongqing, PR China No. cstc2019jcyj-msxmX0338. He wishes to thank the Mathematical Institute of Leiden University.

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