The boundary of hyperbolicity for Henon-like families

Yongluo Cao, Stefano Luzzatto, Isabel Rios

Published 2005-02-11, updated 2007-05-10Version 2

We consider C^{2} Henon-like families of diffeomorphisms of R^{2} and study the boundary of the region of parameter values for which the nonwandering 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 this bifurcation parameter and even, to some extent, at the bifurcation parameter.