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Equilibrium measures for the Hénon map at the first bifurcation

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Published 14 May 2013 • © 2013 IOP Publishing Ltd & London Mathematical Society
, , Citation Samuel Senti and Hiroki Takahasi 2013 Nonlinearity 26 1719 DOI 10.1088/0951-7715/26/6/1719

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senti@im.ufrj.br

Author affiliations

1 Instituto de Matematica, Universidade Federal do Rio de Janeiro, C.P. 68 530, CEP 21941-909, R.J., Brasil

2 Department of Mathematics, Keio University, Yokohama 223-8522, Japan

Dates

  1. Received 11 July 2012
  2. Published

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0951-7715/26/6/1719

Abstract

We study the dynamics of 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 prove the existence of an equilibrium measure which minimizes the free energy associated with the non-continuous potential −t logJu, where $t\in\mathbb R$ is in a certain interval of the form (−∞, t0), t0 > 0 and Ju denotes the Jacobian in the unstable direction.

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Footnotes

  • 3 

    Our arguments and results also hold for Hénon-like families [6, 19], perturbations of the Hénon family.

  • 4 

    We do not make any claim on the continuity of Eu on ∂sR∖{Q}. This does not matter because f-invariant probability measures do not charge this set.

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10.1088/0951-7715/26/6/1719