Symbolic dynamics, shadowing and representation of real numbers with some countably piecewise linear Markov maps

TL;DR

This paper explores countably piecewise linear Markov maps using symbolic dynamics, proving shadowing properties, analyzing rational orbits, and connecting these systems to Diophantine approximation and Cantor series.

Contribution

It introduces $ll$-Markov partitions, relates these maps to Markov shifts, and proves shadowing for a class of systems, extending understanding of their dynamical and number-theoretic properties.

Findings

Proved shadowing property for the studied systems.

Analyzed the orbit structure of rational numbers under these maps.

Provided examples with diverse dynamical behaviors.

Abstract

We study piecewise linear Markov maps, with countable Markov partitions, inspired by a problem of the Mikl\'os Schweitzer competition of the J\'anos Bolyai Mathematical Society in 2022. We introduce $\ell$-Markov partitions and apply ideas of symbolic dynamics to our systems, relating them to Markov shifts. We prove the shadowing property for the system from the competition. We also investigate the possible orbits of rational numbers, for a class of piecewise linear Markov maps which generalize our original system. This has connections with symbolic dynamics, Diophantine approximations and Cantor series. We prove statements on the eventual periodicity of rationals and provide some example systems with different properties.