Dimension of Bernoulli measures for non-linear countable Markov maps

Natalia Jurga (University of Warwick)

Frank Adams 1,

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.

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