Arithmetic Dynamics

TL;DR

This survey introduces Arithmetic Dynamics, a branch of symbolic dynamics focusing on arithmetic expansions of reals and vectors, exploring their properties, codings, and applications in ergodic theory.

Contribution

It provides a comprehensive overview of arithmetic expansions, their dynamical systems, and new insights into their ergodic and combinatorial properties.

Findings

Analysis of beta-expansions and their ergodic properties

Study of rotational and toral expansions in symbolic codings

Development of redundant representations and their combinatorics

Abstract

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.