Recent zbMATH articles in MSC 37B10 https://zbmath.org/atom/cc/37B10 2021-05-28T16:06:00+00:00 Werkzeug Univoque bases of real numbers: local dimension, devil's staircase and isolated points. https://zbmath.org/1459.11022 2021-05-28T16:06:00+00:00 "Kong, Derong" https://zbmath.org/authors/?q=ai:kong.derong "Li, Wenxia" https://zbmath.org/authors/?q=ai:li.wenxia "Lü, Fan" https://zbmath.org/authors/?q=ai:lu.fan "Wang, Zhiqiang" https://zbmath.org/authors/?q=ai:wang.zhi-qiang|wang.zhiqiang "Xu, Jiayi" https://zbmath.org/authors/?q=ai:xu.jiayi Summary: Given a positive integer $$M$$ and a real number $$x>0$$, let $$\mathcal{U}(x)$$ be the set of all bases $$q\in(1,M+1]$$ for which there exists a unique sequence $$(d_i)=d_1d_2\dots$$ with each digit $$d_i\in\{0,1,\dots,M\}$$ satisfying $x=\sum_{i=1}^\infty\frac{d_i}{q^i}.$ The sequence $$(d_i)$$ is called a $$q$$-expansion of $$x$$. In this paper we investigate the local dimension of $$\mathcal{U}(x)$$ and prove a variation principle' for unique non-integer base expansions. We also determine the critical values of $$\mathcal{U}(x)$$ such that when $$x$$ passes the first critical value the set $$\mathcal{U}(x)$$ changes from a set with positive Hausdorff dimension to a countable set, and when $$x$$ passes the second critical value the set $$\mathcal{U}(x)$$ changes from an infinite set to a singleton. Denote by $$\mathbf{U}(x)$$ the set of all unique $$q$$-expansions of $$x$$ for $$q\in\mathcal{U}(x)$$. We give the Hausdorff dimension of $$\mathbf{U}(x)$$ and show that the dimensional function $$x \mapsto\dim_H \mathbf{U}(x)$$ is a non-increasing Devil's staircase. Finally, we investigate the topological structure of $$\mathcal{U}(x)$$. Although the set $$\mathcal{U}(1)$$ has no isolated points, we prove that for typical $$x > 0$$ the set $$\mathcal{U}(x)$$ contains isolated points. Characterization for entropy of shifts of finite type on Cayley trees. https://zbmath.org/1459.37015 2021-05-28T16:06:00+00:00 "Ban, Jung-Chao" https://zbmath.org/authors/?q=ai:ban.jungchao "Chang, Chih-Hung" https://zbmath.org/authors/?q=ai:chang.chih-hung Pseudorandom number generator by $$p$$-adic chaos and Ramanujan expander graphs. https://zbmath.org/1459.05307 2021-05-28T16:06:00+00:00 "Naito, Koichiro" https://zbmath.org/authors/?q=ai:naito.koichiro Summary: In our previous paper [the author, `Randomness of $$p$$-adic discrete dynamical systems and its applications to cryptosystems'', in: Proceedings of the 10th international conference on nonlinear analysis and convex analysis, NACA 2017, Chitose, Japan, 2017. Yokohama: Yokohama Publishers (to appear)], applying chaotic properties of the $$p$$-adic dynamical system given by the $$p$$-adic logistic map, we constructed a new pseudorandom number generator. In this paper, using the pseudorandom sequences given by this generator, we construct random adjacency matrices and their random graphs. Then we numerically show that the eigenvalue distributions of these random matrices have the characteristical properties of the adjacency matrices of Ramanujan graphs. Introduction to hierarchical tiling dynamical systems. https://zbmath.org/1459.37016 2021-05-28T16:06:00+00:00 "Frank, Natalie Priebe" https://zbmath.org/authors/?q=ai:priebe-frank.natalie The paper is devoted to some classical results on dynamical systems arising from substitutions: ordinary (one-dimensional) symbolic systems, one-dimensional tiling systems, multidimensional $$\mathbb{Z}^d$$-systems, and multidimensional tiling systems. The main topics are: various types of substitution systems and relationships between them, dynamical systems arising from these objects, interpretation of standard dynamical properties, super-tiling construction methods, transition matrices, dynamical and diffraction spectra. For the entire collection see [Zbl 1454.37001]. Reviewer: Anton Shutov (Vladimir) Delone sets and dynamical systems. https://zbmath.org/1459.37017 2021-05-28T16:06:00+00:00 "Solomyak, Boris" https://zbmath.org/authors/?q=ai:solomyak.boris The paper is a survey of some (mainly classical) results about substitution Delone sets and some of their generalizations, such as substitution Delone $$m$$-sets, self-affine and pseudo-self-affine tilings. The main focus is on the description of the expansion constants and expansion maps, but some other topics such as connections with dynamical systems and diffraction are also briefly discussed. The paper ends with a short list of open problems on the characterization of expansion maps for tilings with inflation symmetry. For the entire collection see [Zbl 1454.37001]. Reviewer: Anton Shutov (Vladimir) The Ruelle operator for symmetric $$\beta$$-shifts. https://zbmath.org/1459.37029 2021-05-28T16:06:00+00:00 "Lopes, Artur O." https://zbmath.org/authors/?q=ai:lopes.artur-oscar "Vargas, Victor" https://zbmath.org/authors/?q=ai:vargas.victor Summary: Consider $$m \in \mathbb{N}$$ and $$\beta \in (1, m + 1]$$. Assume that $$a \in \mathbb{R}$$ can be represented in base $$\beta$$ using a development in series $$a = \sum^{\infty}_{n = 1}x(n)\beta^{-n}$$, where the sequence $$x = (x(n))_{n \in \mathbb{N}}$$ takes values in the alphabet $$\mathcal{A}_m := \{0, \dotsc, m\}$$. The above expression is called the $$\beta$$-expansion of $$a$$ and it is not necessarily unique. We are interested in sequences $$x = (x(n))_{n \in \mathbb{N}} \in \mathcal{A}_m^\mathbb{N}$$ which are associated to all possible values $$a$$ which have a unique expansion. We denote the set of such $$x$$ (with some more technical restrictions) by $$X_{m,\beta} \subset\mathcal{A}_m^\mathbb{N}$$. The space $$X_{m, \beta}$$ is called the symmetric $$\beta$$-shift associated to the pair $$(m, \beta)$$. It is invariant by the shift map but in general it is not a subshift of finite type. Given a Hölder continuous potential $$A \colon X_{m, \beta} \to\mathbb{R}$$, we consider the Ruelle operator $$\mathcal{L}_A$$ and we show the existence of a positive eigenfunction $$\psi_A$$ and an eigenmeasure $$\rho_A$$ for some values of $$m$$ and $$\beta$$. We also consider a variational principle of pressure. Moreover, we prove that the family of entropies $$(h(\mu_{tA}))_{t > 0}$$ converges, when $$t \to \infty$$, to the maximal value among the set of all possible values of entropy of all $$A$$-maximizing probabilities.