Abundance of $C^1$-robust homoclinic tangencies

Details Abundance of $C^1$-robust homoclinic tangencies Abundance of $C^1$-robust homoclinic tangencies

2009-09-22

arxiv.org/abs/0909.4062v1

Mathematics

Dynamical Systems

39 pages, 11 figures

Scientific paper

A diffeomorphism $f$ has a $C^1$-robust homoclinic tangency if there is a $C^1$-neighbourhood $\cU$ of $f$ such that every diffeomorphism in $g\in \cU$ has a hyperbolic set $\La_g$, depending continuously on $g$, such that the stable and unstable manifolds of $\La_g$ have some non-transverse intersection. For every manifold of dimension greater than or equal to three, we exhibit a local mechanism (blender-horseshoes) generating diffeomorphisms with $C^1$-robust homoclinic tangencies. Using blender-horseshoes, we prove that homoclinic classes of $C^1$-generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display $C^1$-robust homoclinic tangencies.

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