Equilibrium states for S-unimodal maps

Abstract

For S-unimodal maps f, we study equilibrium states maximizing the free energies Ft(μ) := h(μ) - t ∫ log |f′l dμ and the pressure function P(t) := supμ Ft(μ). It is shown that if f is uniformly hyperbolic on periodic orbits, then P(t) is analytic for t ≈ 1. On the other hand, examples are given where no equilibrium states exist, where equilibrium states are not unique and where the notions of equilibrium state for t = 1 and of observable measure do not coincide.