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arXiv · 1412.8012

Equilibrium measures at temperature zero for H\'enon-like maps at the first bifurcation

Hiroki Takahasi
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Abstract

We develop a thermodynamic formalism for a strongly dissipative H\'enon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. For any $t\in\mathbb R$ we prove the existence of an invariant Borel probability measure which minimizes the free energy associated with a non continuous geometric potential $-t\log J^u$, where $J^u$ denotes the Jacobian in the unstable direction. Under a mild condition, we show that any accumulation point of these measures as $t\to+\infty$ minimizes the unstable Lyapunov exponent. We also show that the equilibrium measures converge as $t\to-\infty$ to a Dirac measure which maximizes the unstable Lyapunov exponent.

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Hiroki Takahasi. 2014-12-27. Equilibrium measures at temperature zero for H\'enon-like maps at the first bifurcation. https://arxiv.org/abs/1412.8012

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