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Stable intersections of Cantor sets and homoclinic bifurcations. (English) Zbl 0865.58035
Summary: We study intersections of dynamically defined Cantor sets and consequences to dynamical systems. The concept of stable intersection of two dynamically defined Cantor sets is introduced. We prove that if the stable and unstable Cantor sets associated to a homoclinic bifurcation have a stable intersection, then there are open sets in the parameter line with positive density at the initial bifurcating value, for which the corresponding diffeomorphisms are not hyperbolic. We present conditions more general than the ones previously known that assure stable intersections. We also present conditions for hyperbolicity to be of positive density at homoclinic bifurcations. This allow us to provide persistent one-parameter families of homoclinic bifurcations that present both hyperbolicity and homoclinic tangencies with positive density at the initial bifurcating value in the parameter line.

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
37D99 Dynamical systems with hyperbolic behavior
37E99 Low-dimensional dynamical systems
28A78 Hausdorff and packing measures
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