Dynamics on trees: entropy and substitutions

Abstract

This work has two main parts. In the first, we study Markov tree-shifts given by k transition matrices, one for each of its k directions. We provide a method to characterize the complexity function for these tree-shifts, used to calculate the tree entropies defined by Ban and Chang [4], and Petersen and Salama [23]. Moreover, we compare these definitions of entropy in order to determine some of their properties. The characterization of the complexity function provided is used to calculate the entropy of some examples. Finally, we analyze some topological properties introduced by Ban and Chang [3] for the purpose of answering two of the questions raised by these authors. The second part of this text is dedicated to the investigation of substreetutions. More specifically, we construct a representation of even trees in the tree-shift X defined by Baraviera and Leplaideur [9], in order to obtain a semi-conjugation between Y = {a, b} N×X and the Period Doubling substitution S, a one-dimensional substitutive dynamic. Additionally, we prove that X has uncountably many trees. Afterwards, we dive into characteristics of measures in X, construct a measurable partition for Yev, and give a simple proof of the existence of a measure in Y with maximal entropy. We finish this work investigating the possibility of decomposing X into countably many subsets in such a way that, if a tree belongs to a subset A and one of its preimages is in a subset B, then every tree in B is the preimage of a tree in A.