## 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:
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