Statistical properties of topological Collet-Eckmann maps

We study geometric and statistical properties of complex rational maps satisfying a non-uniform hyperbolicity condition called “Topological Collet-Eckmann”. This condition is weaker than the “Collet-Eckmann” condition. We show that every such map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic, and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and that this invariant measure is exponentially mixing (it has exponential decay of correlations) and satisfies the Central Limit Theorem.We also show that for a complex rational map the existence of such invariant measure characterizes the Topological Collet-Eckmann condition: a rational map satisfies the Topological Collet-Eckmann condition if, and only if, it possesses an exponentially mixing invariant measure that is absolutely continuous with respect to some conformal measure, and whose topological support contains at least 2 points.