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 Poincaré 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.