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

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.