Topological and ergodic properties of symmetric sub-shifts

For a positive integer M and a real base q ∈ (1, M + 1], let U q denote the set of numbers having a unique expansion in base q over the alphabet {0, 1, . . . , M }, and let U q denote the corresponding set of sequences in {0, 1, . . . , M } N . Komornik et al. [Adv. Math. 305 (2017), 165-196] showed recently that the Hausdorff dimension of U q is given by h(U q )/ log q, where h(U q ) denotes the topological entropy of U q . They furthermore showed that the function H : q → h(U q ) is continuous, nondecreasing and locally constant almost everywhere. The plateaus of H were characterized by Alcaraz Barrera et al. [Trans. Amer. Math. Soc., 371 (2019), 3209-3258]. In this article we reinterpret the results of Alcaraz Barrera et al. by introducing a notion of composition of fundamental words, and use this to obtain new information about the structure of the function H. This method furthermore leads to a more streamlined proof of their main theorem.

Section: Introductionmentioning

confidence: 99%

An algebraic approach to entropy plateaus in non-integer base expansions

Allaart¹

2019

“…Formally, we denote by J (H) the set of all points in X whose f -orbit does not intersect H and call it the survivor set. Clearly, a survivor set is f -invariant, and in a number of recent papers certain dynamical properties of the map f | J (H) have been studied -see, e.g., [2] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Open maps: Small and large holes with unusual properties

Hare¹,

Sidorov²

2018

Let X be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in X. We show that there exist arbitrarily small finite overlapping union of shifted cylinders which intersect every orbit under the shift map.We also show that for any proper subshift Y of X there exists a finite overlapping unions of shifted cylinders such that its survivor set contains Y (in particular, it can have entropy arbitrarily close to the entropy of X). Both results may be seen as somewhat counter-intuitive.Finally, we apply these results to a certain class of hyperbolic algebraic automorphisms of a torus.

“…Introduction. In this paper we further investigate the class of open dynamical systems, see [1,3,4,13], that arise from Markov interval maps with non trivial escape set, see [10,11].…”

mentioning

confidence: 99%

“…The first one concerns topological conjugacy where we indeed prove that two Markov interval maps f, g are topological conjugated whenever A f = P A g P T for a certain permutation matrix P , which is in principle unrelated to the permutation that appears in Eq. (1).…”

mentioning

confidence: 99%

Escape dynamics for interval maps

Ramos¹,

Martins²,

Pinto³

2019

We study the structure of the escape orbits for a certain class of interval maps. This structure is encoded in the escape transition matrix A f of an interval map f , extending the traditional matrix A f which considers the transition among the Markov subintervals. We show that the escape transition matrix is a topological conjugacy invariant. We then characterize the 0-1 matrices that can be fabricated as escape transition matrices of Markov interval maps f with escape sets. This shows the richness of this class of interval maps.