2002/03/03 by Nikita Sidorov, Sidorov, Nikita · 5 citations
Computer Science · Mathematics · #11R06 #28D05 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.DS #math.NT #msc:11R06 #msc:28D05 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0203024
45 pages in Latex + 3 figures in eps
arxiv created 2002/03/03 · openalex publication_date 2002/03/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely that they (semi-) conjugate a given continuous (or measure-preserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: (1) Beta-expansions, i.e., the radix expansions in non-integer bases; (2) "Rotational" expansions which arise in the problem of encoding of irrational rotations of the circle; (3) Toral expansions which naturally appear in arithmetic symbolic codings of algebraic toral automorphisms (mostly hyperbolic). We study ergodic-theoretic and probabilistic properties of these expansions and their applications. Besides, in some cases we create "redundant" representations (those whose space of "digits" is a priori larger than necessary) and study their combinatorics.