Two bifurcation sets arising from the beta transformation with a hole at 0
Introduction
Given , the -transformation on the circle is defined by Following the pioneering work of Rényi [10] and Parry [8] there has been a great interest in the study of . In general, the system does not admit a Markov partition (cf. [11]), this makes describing the dynamics of more challenging.
When , Urbański considered in [13], [14] the open dynamical system under the doubling map with a hole at zero. More precisely, for let Here we use a slightly different definition of from that by Urbański. By [13, Theorem 1 and Corollary 1] it follows that the dimension function is a Devil’s staircase on , that is (i) is decreasing and continuous on ; (ii) is locally constant almost everywhere on ; and (iii) is not constant on . Here and throughout the paper denotes the Hausdorff dimension. Moreover, Urbański investigated the bifurcation sets Clearly, . It can be easily deduced from the proof of Theorem 1 in [13] that , and its topological closure 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: for all . Recently, Carminati and Tiozzo in [1] showed that the local Hölder exponent of the dimension function at any equals .
Inspired by the work of Urbański [13], [14], Kalle et al. in [5] considered the analogous problem for the -transformation with a hole . More precisely, for they investigated the survivor set and showed that the dimension function is also a Devil’s staircase on . Furthermore, they characterized the set-valued bifurcation set and proved that is a Lebesgue null set of full Hausdorff dimension for any . Note that the bifurcation set defined here coincides with in [5] (see also Proposition 2.3(i)). Interestingly, they showed that contains infinitely many isolated points for Lebesgue almost every . This is in contrast to the case where and has no isolated points. For -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 the dimension function is a Devil’s staircase, it is natural to consider the dimension bifurcation set This set records those for which the dimension function has a ‘change’ within any right neighborhood. Since is continuous, cannot have isolated points. On the other hand, the set-valued bifurcation set contains (infinitely many) isolated points for Lebesgue almost every . So in general we cannot expect the coincidence of the two bifurcation sets and . That being said, in this paper we show that if is a multinacci number, i.e., the unique root in of the equation for some , then the two bifurcation sets indeed coincide. Importantly, if is a multinacci number then its quasi-greedy expansion of is of the form . This property will be useful in our analysis. Here for the quasi-greedy -expansion of is the lexicographically largest zero–one sequence not ending with an infinite string of zeros and satisfying (see Section 2 for more details). Furthermore, throughout the paper we will use lexicographical order ‘’ and ‘’ between sequences and words.
When is a multinacci number, the following result for the set-valued bifurcation set was established in [5, Theorems C and D]. We record it here for later use.
Theorem 1.1 [5] Let be a multinacci number. Then the topological closure is a Cantor set. Furthermore, .
In order to give a complete description of the dimension bifurcation set we introduce a class of basic intervals.
Definition 1.2 Let . A word is called -Lyndon if Accordingly, an interval is called a -Lyndon interval if there exists a -Lyndon word such that
Here we mention that in Definition 1.2 the left endpoint has a finite -expansion and the right endpoint has a periodic -expansion, see Section 2 for more explanations.
We will show that the -Lyndon intervals are pairwise disjoint for all , and when is multinacci they cover the interval 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 be a multinacci number. Then where the union is taken over all pairwise disjoint -Lyndon intervals.
By Theorem 1 it follows that the topological closure of each -Lyndon interval is indeed a maximal interval where the dimension function is constant. As a corollary of Theorem 1 we confirm a conjecture of [5] for a multinacci number.
Corollary 2 If is a multinacci number, then
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 . 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 .
Section snippets
Preliminaries and -Lyndon intervals
Given , for each there exists a sequence such that The sequence is called a -expansion of . Sidorov [12] showed that for Lebesgue almost every has a continuum of -expansions. This is rather different from the case when where every number in has a unique dyadic expansion except for countably many points that have precisely two expansions. Given , among all of its -expansions let be
Proof of Theorem 1
In this section we will prove Theorem 1. First we show that the dimension bifurcation set coincides with the set-valued bifurcation set , we then derive a complete characterization of these sets via the -Lyndon intervals. The proof heavily relies upon the transitivity of the symbolic survivor set (see Lemma 3.2).
Proposition 3.1 Let be a multinacci number. Then where the union is taken over all -Lyndon intervals.
Observe by Lemma 2.2 that the -Lyndon intervals are
Final remarks
The main results obtained in this paper can be easily modified to study the following analogous bifurcation sets: If is a multinacci number, one can show that where the union is taken over all pairwise disjoint closed -Lyndon intervals.
Observe that the main result Theorem 1 holds under the assumption that 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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