EPFL
Lausanne
Contribution to the ergodic theory of piecewise monotone continuous maps
Faller
Bastien
2008
This thesis is devoted to the ergodic theory of the piecewise monotone continuous maps of the interval. The coding is a classical approach for these maps. Thanks to the coding, we get a symbolic dynamical system which is almost isomorphic to the initial dynamical system. The principle of the coding is very similar to the one of expansion of real numbers. We first define the coding in a perspective similar to the one of the expansions of real numbers; this perspective was already adopted by Rényi and Parry in their papers about the expansions of numbers. Then we present the theory of Hofbauer about the links between the ergodic properties of a piecewise monotone continuous map of the interval and the corresponding symbolic dynamical system. We prove that there is a bijection between the sets of measures of maximal entropy of these two dynamical systems. We apply these results to the study of two families of maps: first the maps Tα,β(x) := βx + α mod 1, then the maps we will call generalized β-transformations. For the family of maps Tα,β, we describe in detail the family of symbolic dynamical systems obtained by the coding. Then we turn to the question of normality of the orbits for the maps Tα,β. Finally we study the generalized β-transformations: we prove that most of them have a a unique measure of maximal entropy, then we also study the normality of the orbits for theses maps.
technical-report