# Geometric Rigidity in Conformal Dynamical Systems

4:00 pm Thursday, September 26, 2013
Mariusz Urbanski (University of North Texas)

I will discuss the theorem which asserts that if $$G$$ is a Kleinian group acting on a finite-dimensional hyperbolic space, then under the assumption that the Hausdorff dimension of the complement of the radial limit set in the limit set of $$G$$ is smaller than the Poincare exponent of $$G$$, then either the Hausdorff dimension of the limit set of $$G$$ is larger than its topological dimension $$k$$ or this limit set is a geometric sphere of dimension $$k$$. This extends the result of M. Kapovich [2009] from the case of geometrically finite Kleinian groups. The related theorems for conformal iterated function systems and rational functions of the Riemann sphere also will be discussed. The case of infinite dimension, Hilbert space, also will be mentioned.