### 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 noncontinuous geometric potential -t log J_{u}, where t 2 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.

Original language | English |
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Journal | Ergodic Theory and Dynamical Systems |

Volume | 760 |

DOIs | |

Publication status | Published - 2014 Nov 17 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Equilibrium measures for the Hénon map at the first bifurcation : Uniqueness and geometric/statistical properties.** / Senti, Samuel; Takahasi, Hiroki.

Research output: Contribution to journal › Article

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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

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.

UR - http://www.scopus.com/inward/record.url?scp=84910671527&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84910671527&partnerID=8YFLogxK

U2 - 10.1017/etds.2014.61

DO - 10.1017/etds.2014.61

M3 - Article

AN - SCOPUS:84910671527

VL - 760

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

ER -