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We consider the dynamics of strongly dissipative H\'enon-like maps in theplane, around the first bifurcation parameter $a^*$ at which the uniformhyperbolicity is destroyed by the formation of homoclinic or heteroclinictangencies inside the limit set. In [Takahasi H.: Commun. Math. Phys. {\bf 312}37-85 (2012)], it was proved that $a^*$ is a full Lebesgue density point of theset of parameters for which Lebesgue almost every initial point diverges toinfinity under forward iteration. For these parameters, we show that allLyapunov exponents of all invariant ergodic Borel probability measures areuniformly bounded away from zero, uniformly over all the parameters.

Prevalence of non-uniform hyperbolicity at the first bifurcation of H\'enon-like families