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Equilibrium stability for non-uniformly hyperbolic systems

Published online by Cambridge University Press:  18 January 2018

JOSÉ F. ALVES
Affiliation:
Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal email [email protected]
VANESSA RAMOS
Affiliation:
Centro de Ciências Exatas e Tecnologia-UFMA, Av. dos Portugueses, 1966, Bacanga, 65080-805 São Luís, Brazil email [email protected]
JAQUELINE SIQUEIRA
Affiliation:
Departamento de Matemática PUC-Rio, Marquês de Sâo Vicente 225, Gávea, 225453-900 Rio de Janeiro, Brazil email [email protected]

Abstract

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials. Finally, we show that these equilibrium states vary continuously in the $\text{weak}^{\ast }$ topology within such systems.

Type
Original Article
Copyright
© Cambridge University Press, 2018 

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