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

We study piecewise linear and piecewise Lipschitz 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 investigate the possible orbits of rational numbers for a class of piecewise linear Markov maps that 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. We prove the shadowing property for a class of systems on a noncompact domain that are piecewise Lipschitz and uniformly expanding, including the system from the competition.