Iteration of rational functions: inducing, nice and pleasant couples

I plan to sketch results of my recent paper joint with Juan Rivera-Letelier, concerning pressure function $P(t)$ of potential $-t\log |f|$ for weakly hyperbolic rational mapping $f$. The results concern the existence of conformal and invariant equilibrium states, analiticity of the pressure, analyticity of dimension spectrum of Lyapunov exponents, etc. The class of mappings includes all finitely renormalizable polynomials without parabolic periodic orbits, infinitely renormalizable polynomials with complex a priori bounds, Topological Collet-Eckmann maps.