Abstract

We are interested in discrete dynamical systems on the interval, that is, the iteration of a continuous map $f:[0,1]\; -->\; [0,1]$, and we look for invariant measures of maximal entropy. For this purpose, we associate a topological Markov chain (i.e., a symbolic system on an infinite graph) to the interval map. I will illustrate this construction on a example and I will explain why the topological Markov chains are a suitable tool when looking for measures of maximal entropy. This technique was first introduced by Hofbauer, who proved that any piecewise monotone map has a measure of maximal entropy, which is unique in the transitive case, then Buzzi showed that the same result holds for interval maps. However there exist transitive $Cn$ maps with no measure of maximal entropy, for arbitrarily large $n$. Buzzi and I showed a sufficient condition for existence of measure of maximal entropy for $C1$ interval maps.