Abstract. We study the dynamics of 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 prove the existence of an equilibrium measure which minimizes the free energy associated with the non continuous potential −t log J u , where t ∈ R is in a certain interval of the form (−∞, t 0 ), t 0 > 0 and J u denotes the Jacobian in the unstable direction.

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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 ∈ 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.

We study the dynamics of strongly dissipative Hénon-like maps, around the first bifurcation parameter a * at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove that a * is a full Lebesgue density point of the set of parameters for which Lebesgue almost every initial point diverges to infinity under positive iteration. A key ingredient is that a * corresponds to the "non-recurrence of every critical point", reminiscent of Misiurewicz parameters in one-dimensional dynamics. Adapting on the one hand Benedicks & Carleson's parameter exclusion argument, we construct a set of "good parameters" having a * as a full density point. Adapting Benedicks & Viana's volume control argument on the other, we analyze Lebesgue typical dynamics corresponding to these good parameters.

For a certain parametrized family of maps on the circle with critical points and logarithmic singularities where derivatives blow up to infinity, we construct a positive measure set of parameters corresponding to maps which exhibit nonuniformly expanding behavior. This implies the existence of "chaotic" dynamics in dissipative homoclinic tangles in periodically perturbed differential equations.