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Title:
Mini-course. Richness of dynamics near homoclinic tangencies Part II
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Abstract:
Given a difeomorphism F of an m-dimensional space M we can characterize its behavior on small spatial scales by means of the following procedure. Take a small ball g(B) where B is a unit n-dimensional ball and g is a diffeomorphism acting from some neighborhood U of B into M. Let h be the inverse of g. If for some integer k > 0 the image of g (B) by the k-th iteration F^k of F lies in g(U), we can consider a renormalized iteration h F^k g which is a diffeomorphism of B onto its image in U. The set of all renormalized iterations (for all possible maps g and numbers k) is called the renormalization class of F. We say that dynamics of F are rich if this class is large. We show that by a small perturbation of any two-dimensional map with a homoclinic tangency, one can obtain dynamics of ultimate richness. Namely, if the map is area-contracting, the closure of the renormalization class contains all one-dimensional dynamics; in the area-preserving case it contains all conservative dynamics; if contraction of areas coexists with area-expansion, then the closure of the renormalized class contains all two-dimensional dynamics. We continue by showing how these results can be carried out for dimension 3 and higher. The main tool is the renormalization to Henon-like maps.
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