The boundary of hyperbolicity for Henon-like families

Abstract

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.

32 pages, 11 figures. Several minor revisions, additional figures, clarifications of some arguments

References in corpus (2)

• Parameter exclusions in Henon-like systems
• Stable manifolds under very weak hyperbolicity conditions

Cited by in corpus (6)

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• A Symbolic Characterization of the Horseshoe Locus in the Hénon Family
• Removal of phase transition of the Chebyshev quadratic and thermodynamics of Hénon-like maps near the first bifurcation