Seminar

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

#### Fractal Geometry and Dynamics

#### 23 November 15:00 - 15:50

#### Natalia Jurga - University of Surrey

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.

Organizers

Kenneth Falconer

University of St Andrews

Maarit Järvenpää

University of Oulu

Antti Kupiainen

University of Helsinki

Francois Ledrappier

University of Notre Dame

Pertti Mattila

University of Helsinki