Expanding in the Mean

Abstract

In this chapter we show that the main achievements of this manuscript, including thermodynamical formalism, Bowen’s formula and multifractal analysis, also hold for a class of random maps satisfying an allegedly weaker expanding condition

$$\int \nolimits \nolimits \log {\gamma }_{x}\mathit{dm}(x) > 0.$$

We start with a precise definition of this class. Then we explain how this case can be reduced to random expanding maps by looking at an appropriate induced map. The picture is completed by providing and discussing a concrete map that is not expanding but expanding in the mean.