Gometric Exponents for Hyperbolic Julia Sets

We show that the Hausdorff dimension of the Julia set associated to a hyperbolic rational map is bounded away from 2, where the bound depends only on certain intrinsic geometric exponents. This result is derived via lower estimates for the iterate-counting function and for the dynamical Poincare series. We deduce some interesting consequences, such as upper bounds for the decay of the area of parallel-neighbourhoods of the Julia set, and lower bounds for the Lyapunov exponents with respect to the measure of maximal entropy.