Chaos in topological dynamics, in particular on the interval, measures of maximal entropy

Summary of the Ph.D. thesis

In this thesis we are interested in the properties linked to chaos and in the existence of measures of maximal entropy (called maximal measures) for some systems, in particular for systems on the interval.

For a topological dynamical system $(X,T)$, having a positive entropy is considered as a chaotic property. We show that a positive topological entropy implies the existence of proper asymptotic pairs, that is, pairs of distinct points $(x,y)$ such that the distance between $T^n x$ and $T^n y$ tends to zero when n goes to infinity. In addition, if $T$ is invertible, many asymptotic pairs for T are Li-Yorke pairs for the inverse of $T$. The proofs of these results rely entirely on ergodic methods.

A topological Markov chain is the set of infinite paths on an oriented graph; these systems are a tool for the study of maximal measures. A connected oriented graph is either transient or null recurrent or positive recurrent. We recall the links between this classification and the fact that a graph can be extended or contracted without changing its entropy, and we show that any transient graph is included in a recurrent graph of equal entropy. It is known that a Markov chain on a connected graph $G$ has a maximal measure if and only if $G$ is positive recurrent. We give a new condition implying positive recurrence and we show the existence of almost maximal measures escaping to infinity for non positive recurrent graphs.

When one restricts to dynamical systems on the interval, the various notions of chaos mostly coincide. We survey the links between the different chaotic properties.

For an interval map, the question of existence of maximal measures reduces in some cases to the study of a Markov chain. This allows us to give a condition that ensures the existence of a maximal measure for $C^1$ maps. For every integer $n$, we build an example of a $C^n$ mixing interval map which has no maximal measure.

Thesis: [ps.gz] [pdf]