The boundary of hyperbolicity for Hénon-like families

Abstract

AbstractWe consider C2 Hénon-like families of diffeomorphisms of $ \mathbb R^{2} $ and study the boundary of the region of parameter values for which the non-wandering set is uniformly hyperbolic. Assuming sufficient dissipativity, we show that the loss of hyperbolicity is caused by a first homoclinic or heteroclinic tangency and that uniform hyperbolicity estimates hold uniformly in the parameter up to the bifurcation parameter and even, to some extent, at the bifurcation parameter.