Equilibrium measures at temperature zero 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 noncontinuous geometric potential -t log J^u, where J^u denotes the Jacobian in the unstable direction. We characterize accumulation points of these measures as t→±∞ in terms of the unstable Lyapunov exponent.