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Equilibrium states for S-unimodal maps

Published online by Cambridge University Press:  01 August 1998

HENK BRUIN
Affiliation:
Mathematisches Institut, Universität Erlangen-Nürnberg D-91054 Erlangen, Germany
GERHARD KELLER
Affiliation:
Mathematisches Institut, Universität Erlangen-Nürnberg D-91054 Erlangen, Germany

Abstract

For S-unimodal maps $f$, we study 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 $t\approx 1$. 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.

Type
Research Article
Copyright
© 1998 Cambridge University Press

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