## Dimension of Bernoulli measures for non-linear countable Markov maps

#### Natalia Jurga (University of Warwick)

It is well known that the Gauss map $$G: [0,1) \to [0,1)$$

$$G(x)= \frac{1}{x} \mod 1$$

has an absolutely continuous invariant probability measure $$\mu_G$$ given by

$$\mu_G(A)= \frac{1}{\log 2} \int_A \frac{1}{1+x} dx$$

Kifer, Peres and Weiss showed that there exists a `dimension gap' between the supremum of the Hausdorff dimensions of Bernoulli measures $$\mu_{\mathbf{p}}$$ for the Gauss map and the dimension of the measure of maximal dimension (which in this case is $$\mu_G$$ with dimension 1). In particular they showed that

$$\sup_{\mathbf{p}} \dim_H \mu_{\mathbf{p}} < 1- 10^{-7}$$

In this talk we consider the geometric properties of $$T$$ which lead to a dimension gap.