# zbMATH — the first resource for mathematics

Equilibrium states for $$S$$-unimodal maps. (English) Zbl 0916.58020
The authors study for $$S$$-unimodal maps $$f$$ equilibrium states maximizing the free energies $$F_t(\mu):= h(\mu)+ t\int\log| f'| d\mu$$ and the pressure function $$P(t):= \sup_\mu F_t(\mu)$$. It is shown that if $$f$$ is uniformly hyperbolic on periodic orbits, then $$P(t)$$ is analytic for $$t\approx 1$$. Morover, the authors investigate the stability of $$F_t$$ for a large class of functions but also give an example of a logistic map that is not stable and has no equilibrium state.

##### MSC:
 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
Full Text: