Dimension of Bernoulli measures for non-linear countable Markov maps

Fractal Geometry and Dynamics

23 November 15:00 - 15:50

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.
Kenneth J. 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


Maarit Järvenpää


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