Download PDF by Christian Bonatti, Lorenzo J. Díaz, Marcelo Viana: Dynamics Beyond Uniform Hyperbolicity: A Global Geometric

By Christian Bonatti, Lorenzo J. Díaz, Marcelo Viana

ISBN-10: 3540220666

ISBN-13: 9783540220664

ISBN-10: 3540268448

ISBN-13: 9783540268444

In large phrases, the objective of dynamics is to explain the long term evolution of structures for which an "infinitesimal" evolution rule, reminiscent of a differential equation or the new release of a map, is understood.

The inspiration of uniform hyperbolicity, brought through Steve Smale within the early sixties, unified vital advancements and ended in a remarkably profitable conception for a wide classification of platforms: uniformly hyperbolic structures frequently express advanced evolution which, however, is now very well understood, either geometrically and statistically.

Another revolution has been occurring within the final couple of many years, as one attempts to construct an international conception for "most" dynamical platforms, recuperating up to attainable of the conclusions of the uniformly hyperbolic case, in nice generality.

This e-book goals to place such fresh advancements in a unified standpoint, and to indicate open difficulties and sure instructions for extra growth. it truly is geared toward researchers, either younger and senior, prepared to get a brief, but vast, view of this a part of dynamics. major rules, equipment, and effects are mentioned, at variable levels of intensity, with references to the unique works for info and complementary information.

The 12 chapters are organised on the way to express an international point of view of this box, yet they've been saved fairly self sustaining, to permit direct entry to precise themes. The 5 appendices conceal very important complementary fabric.

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Additional resources for Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective

Example text

In the simplest case H is just the orbit of p. We take all maps to be at least C2. Moreover, for the time being we suppose that the ambient manifold M is a surface. This is crucial to ensure that the families W s = {Wfoc(x) : x e H} and Wu = {W^x) : x G H} of local stable manifolds and local unstable manifolds extend to C1 foliations T8 and Tu denned on a neighborhood of H. In fact, the extensions may be chosen so that the tangent bundles of Ts and Tu are generated by C1 vector fields. See [345, Appendix 1].

2 Persistent tangencies, coexistence of attractors It was a remarkable discovery of Newhouse [319] that the Cantor sets Ks and Ku may intersect each other in a robust fashion, that is for a whole open set of diffeomorphisms close to the original / . Here openness is relative to x This fact, together with the local product property, implies that the hyperbolic set H is locally C1 -diffeomorphic to a product (Wfc^x) D H) x (Wfoc(x) Pi H) close to any point x £ H. 28 3 Homoclinic Tangencies the C2 topology, for reasons that will become apparent from the explanation that follows.

In addition, the complement has zero Lebesgue measure: more than that, HD^if 1 — K2) \ S^if^if 2 )) is less than 1. Let (/M)M be a smooth family of surface diffeomorphisms generically unfolding a homoclinic tangency associated to a hyperbolic set H of / = /Q. 19 and using results from [313], Moreira, Yoccoz show that, if the Hausdorff dimension of H is larger than 1, then persistent tangencies correspond to a definite fraction of parameters close to zero and the union of such parameters with those corresponding to hyperbolicity has full Lebesgue density at the bifurcation.

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Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective by Christian Bonatti, Lorenzo J. Díaz, Marcelo Viana


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