Sets of periods of dynamical systems

In this article, we present a coherent, though not exhaustive, account of some well-known and some recent results of many mathematicians (including our own) on the following question: Given a "nice" class of dynamical systems, which subsets of N arise as the sets of periods of members of that class? While stating and explaining some elegant answers, proofs have been outlined or indicated occasionally

Topological properties of cellular automata on trees

We prove that there do not exist positively expansive cellular automata defined on the full k-ary tree shift (for k>=2). Moreover, we investigate some topological properties of these automata and their relationships, namely permutivity, surjectivity, preinjectivity, right-closingness and openness.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

Integrable and Chaotic Systems Associated with Fractal Groups

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 80-s of the last century with the purpose to solve some famous problems in mathematics, including the question raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor's question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schr\"odinger operators. One of important developments is the relation of them to the multi-dimensional dynamics, theory of joint spectrum of pencil of operators, and spectral theory of Laplace operator on graphs. The paper gives a quick access to these topics, provide calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains discussion of the dichotomy "integrable-chaotic" in the considered model, and suggests a possible probabilistic approach to the study of discussed problems.Comment: 48 pages, 15 figure

A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications

Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation. Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modelling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been introduced in update pattern, lattice structure, neighborhood dependency and local rule. In this survey, we tour to the various types of CAs introduced till date, the different characterization tools, the global behaviors of CAs, like universality, reversibility, dynamics etc. Special attention is given to non-uniformity in CAs and especially to non-uniform elementary CAs, which have been very useful in solving several real-life problems.Comment: 43 pages; Under review in Natural Computin

Equilibrium states for non-transitive random open and closed dynamical systems

This work provides random Perron--Frobenius decompositions and relative equilibrium states for a class of random open and closed interval maps, without imposing transitivity requirements, such as mixing and covering conditions, which are prevalent in the literature. Such decompositions are linked to the existence and uniqueness of random conformal and invariant measures with exponential decay of correlations. These results allow us to expand the class of examples of (random) dynamical systems amenable to multiplicative ergodic theory and thermodynamic formalism. Applications include open and closed random intermittent maps with geometric potentials, non-transitive random maps and a connection between Lyapunov exponents and escape rates through random holes

On Undecidable Dynamical Properties of Reversible One-Dimensional Cellular Automata

Cellular automata are models for massively parallel computation. A cellular automaton consists of cells which are arranged in some kind of regular lattice and a local update rule which updates the state of each cell according to the states of the cell's neighbors on each step of the computation. This work focuses on reversible one-dimensional cellular automata in which the cells are arranged in a two-way in_nite line and the computation is reversible, that is, the previous states of the cells can be derived from the current ones. In this work it is shown that several properties of reversible one-dimensional cellular automata are algorithmically undecidable, that is, there exists no algorithm that would tell whether a given cellular automaton has the property or not. It is shown that the tiling problem of Wang tiles remains undecidable even in some very restricted special cases. It follows that it is undecidable whether some given states will always appear in computations by the given cellular automaton. It also follows that a weaker form of expansivity, which is a concept of dynamical systems, is an undecidable property for reversible one-dimensional cellular automata. It is shown that several properties of dynamical systems are undecidable for reversible one-dimensional cellular automata. It shown that sensitivity to initial conditions and topological mixing are undecidable properties. Furthermore, non-sensitive and mixing cellular automata are recursively inseparable. It follows that also chaotic behavior is an undecidable property for reversible one-dimensional cellular automata.Siirretty Doriast