Abstract:

We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most log 2, and the metric entropy is not larger than the topological entropy.

In order to have a better control on the metric entropy, we work mainly with topologically mixing piecewise expanding skew tent maps, for which there are only 2 diff erent slopes. For those maps, there is an additional restriction that the topological entropy is larger than (1/2) log 2.

Moreover, we generalize and give a diff erent interpretation of the Milnor-Thurston formula connecting the topological entropy and the kneading determinant for unimodal maps.

This is joint work with Lluís Alsedà and Rodrigo Pérez.