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arXiv · 1209.2224

Equilibrium measures for the H\'enon map at the first bifurcation: uniqueness and geometric/statistical properties

Samuel Senti·Hiroki Takahasi
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Abstract

For strongly dissipative H\'enon maps at the first bifurcation where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e., prove the existence and uniqueness of an invariant probability measure which maximizes 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 $J^u$ is the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

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Samuel Senti, Hiroki Takahasi. 2012-09-11. Equilibrium measures for the H\'enon map at the first bifurcation: uniqueness and geometric/statistical properties. https://doi.org/10.1017/etds.2014.61

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