ABSTRACT: Let $f:\mathbb{C}\to \mathbb{C}$ be an arbitrary
holomorphic endomorphism of degree larger than $1$ of the Riemann
sphere $\mathbb{C}$.. Denote by $J(f)$ its Julia set. Let $\varphi:J(f)
\to\mathbb{R}$ be a H\"older continuous function whose topological
pressure exceeds its supremum. It is known that then
there exists a unique equilibrium measure $\mu_\varphi$ for this
potential. I will discuss a special inducing scheme with fine
recurrence properties. This construction allows us to prove three
results.

Dimension rigidity, i.e. a characterization of all maps and
potentials for which $HD(\mu_\varphi)=HD(J(f))$.

Real analyticity of topological pressure $P(t\varphi)$ as a function of
$t$.

Exponential decay of correlations, and, as its consequence, the Central
Limit Theorem and the Law of Iterated Logarithm for H\"older
continuous observables. Finally, the Law of Iterated Logarithm for all
linear combinations of H\"older continuous observables and the
function $\log|f'|$.

Geometric consequences of the Law of Iterated Logarithm lead to
comparison of equilibrium states with appropriately generalized
Hausdorff measures on the Julia set $J(f)$.