Entropy and recurrent dimensions of discrete dynamical systems given by p-adic expansions

Abstract

In this paper we study symbolic dynamical systems given by the shift mapping on the coefficient sequences of expansions of p-adic numbers. We associate the upper and lower recurrent dimensions with the topological entropies of these discrete dynamical systems by giving some inequalities representing the relationships among these parameters. Using these inequality relations, we estimate the topological entropies or recurrent dimensions of the various coefficient sequences, which have recurrent properties. For the case of Sturmian sequences we can estimate the positive gap values between upper and lower recurrent dimensions, which indicate the unpredictability of the orbits, if the irrational frequencies of the sequences are Liouville numbers.