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Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties

Published online by Cambridge University Press:  02 October 2014

SAMUEL SENTI
Affiliation:
Instituto de Matematica, Universidade Federal do Rio de Janeiro, C.P. 68 530, CEP 21945-970, R.J., Brasil email senti@im.ufrj.br
HIROKI TAKAHASI
Affiliation:
Department of Mathematics, Keio University, Yokohama 223-8522, Japan email hiroki@math.keio.ac.jp

Abstract

For strongly dissipative Hénon maps at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e. we prove the existence and uniqueness of an invariant probability measure that minimizes the free energy associated with a non-continuous geometric potential $-t\log J^{u}$, where $t\in \mathbb{R}$ is in a certain large interval and $J^{u}$ denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

Type
Research Article
Copyright
© Cambridge University Press, 2014 

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