TY - JOUR

T1 - Equilibrium measures for the Hénon map at the first bifurcation

T2 - Uniqueness and geometric/statistical properties

AU - Senti, Samuel

AU - Takahasi, Hiroki

N1 - Funding Information:
S.S. is partially supported by the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq) and PRONEX, Brazil. H.T. is partially supported by the Grant-in-Aid for Young Scientists (B) of the Japan Society for the Promotion of Science (JSPS), Grant No. 2374012, and the Keio Gijuku Academic Development Funds. This research is partially supported by the Kyoto University Global COE (Centers of Excellence) Program. We thank the Mathematics Departments of Kyoto University, the Federal University of Rio de Janeiro, Pennsylvania State University, l'Ecole Polytechnique Federale de Lausanne, and IMPA (Instituto Nacional de Matematica Pura e Aplicada) for their hospitality.
Publisher Copyright:
© Cambridge University Press, 2014.

PY - 2014/11/17

Y1 - 2014/11/17

N2 - 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 noncontinuous geometric potential -t log Ju, where t 2 R is in a certain large interval and Ju denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

AB - 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 noncontinuous geometric potential -t log Ju, where t 2 R is in a certain large interval and Ju denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

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U2 - 10.1017/etds.2014.61

DO - 10.1017/etds.2014.61

M3 - Article

AN - SCOPUS:84951765841

VL - 760

SP - 215

EP - 255

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

ER -