Natalia Jurga

Dimension of Bernoulli measures for non-linear countable Markov maps

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
$$\sup_{\mathbf{p}} \dim_H \mu_{\mathbf{p}} < 1- 10^{-7}$$
They also proved analogous results for maps $T$ arising from $f$-expansions with a corresponding absolutely continuous measure $\mu_T$, under the condition that the digits of the $f$-expansion were dependent with respect to $\mu_T$. However, sometimes the absolutely continuous measure is not known or the above condition is difficult to verify. Instead, we consider the underlying geometric cause of the dimension gap; in particular we show that under an explicit non-linearity condition on the map $T$ we obtain a dimension gap.