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

2025/08/23 by Kalocsai, Zoltán · 1 citation
#37A50 #37B65 #37E05 #Dynamical Systems (math.DS) #FOS: Mathematics #Primary: 37B10 Secondary: 11K55

paper · doi:10.48550/arxiv.2508.17159

Abstract

We study piecewise linear Markov maps, with countable Markov partitions, inspired by a problem of the Miklós Schweitzer competition of the János Bolyai Mathematical Society in 2022. We introduce ℓ-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.

Citations

• Convergence of pushforward measures for certain countably piecewise linear Markov maps
• Symbolic Dynamics and Markov Partitions
• Representations for real numbers and their ergodic properties
• Criteria for Irrationality of Certain Classes of Numbers
• Pseudo-orbit shadowing in the family of tent maps

Cited by

• Convergence of pushforward measures for certain countably piecewise linear Markov maps

Related

• Symbolic Dynamics and Markov Partitions
• Convergence of pushforward measures for certain countably piecewise linear Markov maps
• Finitely presented dynamical systems
• Persistent Markov partitions for rational maps
• Persistent Markov partitions and hyperbolic components of rational maps