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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
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