arXiv subjects

Rafael Alcaraz Barrera

Publications and source records attributed to Rafael Alcaraz Barrera.

7 recordsLinked to original sources

arXiv2025

Topological and dimensional properties of univoque bases in double-base expansions

Given two real numbers $q_0,q_1$ with $q_0, q_1 > 1$ satisfying $q_0+q_1 \ge q_0q_1$, we call a sequence $(d_i)$ with $d_i\in \{0,1\}$ a $(q_0,q_1)$-expansion or a double-base expansion of a real number $x$ if \[ x=\mathop{\sum}\limits_{i=1}^{\infty} \frac{d_{i}}{q_{d_1}q_{d_2}\cdots q_{d_i}}. \] When $q_0=q_1=q$, the set of univoque bases is given by the set of $q$'s such that $x = 1$ has exactly one $(q, q)$-expansion. The topological, dimensional and symbolic properties of such sets and their corresponding sequences have been intensively investigated. In our research, we study the topological and dimensional properties of the set of univoque bases for double-base expansions. This problem is more complicated, requiring new research strategies. Several new properties are uncovered. In particular, we show that the set of univoque bases in the double base setting is a meagre set with full Hausdorff dimension.

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arXiv2022

Chaotic sets and Hausdorff dimension for Lüroth expansions

We provide new similarities between regular continued fractions and Lüroth series in terms of topological dynamics and Hausdorff dimension. In particular, we establish a complete analogue for the Lüroth transformation of results by W. Liu, B. Li and W. Liu, S. Wang on the distal, asymptotic and Li-Yorke pairs for the Gauss map.

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arXiv2020

On the specification property and synchronisation of unique $q$-expansions

Given a positive integer $M$ and $q \in (1, M+1]$ we consider expansions in base $q$ for real numbers $x \in \left[0, {M}/{q-1}\right]$ over the alphabet $\{0, \ldots, M\}$. In particular, we study some dynamical properties of the natural occurring subshift $(\mathbf{V}_q, σ)$ related to unique expansions in such base $q$. We characterise the set of $q \in (1,M+1]$ such that $(\mathbf{V}_q, σ)$ has the specification property and the set of $q \in (1,M+1]$ such that $(\mathbf{V}_q, σ)$ is a synchronised subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of $q$. We also calculate the size of such classes giving similar results to those shown by Schmeling in (Ergodic Theory and Dynamical Systems, 17:675--694, 6 1997) in the context of $β$-transformations.

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arXiv2017

Entropy, topological transitivity, and dimensional properties of unique $q$-expansions

Let $M$ be a positive integer and $q \in(1,M+1].$ We consider expansions of real numbers in base $q$ over the alphabet $\{0,\ldots, M\}$. In particular, we study the set $\mathcal{U}_{q}$ of real numbers with a unique $q$-expansion, and the set $\mathbf{U}_q$ of corresponding sequences. It was shown in (Komornik et al, 2017 Adv. Math.) that the function $H$, which associates to each $q\in(1, M+1]$ the topological entropy of $\mathcal{U}_q$, is a Devil's staircase. In this paper we explicitly determine the plateaus of $H$, and characterize the bifurcation set $\mathcal E$ of $q$'s where the function $H$ is not locally constant. Moreover, we show that $\mathcal E$ is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift $(\mathbf{V}_q, σ),$ which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of $\mathcal{U}_q$ coincide for all $q\in(1,M+1]$.

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arXiv2015

Intrinsic Ergodicity of Open dynamical systems for the doubling map

We give sufficient conditions for intervals $(a,b)$ such that the associated open dynamical system for the doubling map is intrinsically ergodic. We also show that the set of parameters $(a,b) \in (\frac{1}{4}, \frac{1}{2}) \times (\frac{1}{2},\frac{3}{4})$ such that the attractor $(Λ_{(a,b)}, f_{(a,b)})$ is intrinsically ergodic has full Lebesgue measure and we construct a set of points where intrinsic ergodicity does not hold.

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arXiv2015

Topological dynamics of the doubling map with asymmetrical holes

We study the dynamics of the attractor of the doubling map with an asymmetrical hole by associating to each hole an element of the lexicographic world. A description of the topological entropy function is given. We show that the set of parameters $(a,b)$ such that the dynamics of the mentioned attractor corresponds to a subshift of finite type is open and dense. Using the connections between this family of open dynamical systems, intermediate $β$-expansions and Lorenz maps we study the topological transitivity and the specification property for such maps.

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arXiv2013

Topological and Ergodic properties of symmetric subshifts

The family of symmetric one sided subshifts in two symbols given by a sequence $a$ is studied. We analyse some of their topological properties such as transitivity, the specification property and intrinsic ergodicity. It is shown that almost every member of this family admits only one measure of maximal entropy.

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