Lyapunov Optimizing Measures for Hénon-like Maps at the First Bifurcation

We develop a thermodynamic formalism for a strongly dissipative Hénon-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 ∈ 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, where J denotes the Jacobian in the unstable direction. Under a mild condition, we show that any accumulation point of these measures as t → +∞ is a measure which minimizes the unstable Lyapunov exponent. We also show that the equilibrium measures converge as t → −∞ to a Dirac measure which maximizes the unstable Lyapunov exponent. This is an excerpt from the paper [20].