By Christian Bonatti, Lorenzo J. Díaz, Marcelo Viana

The suggestion of uniform hyperbolicity, brought by means of Steve Smale within the early sixties, unified very important advancements, and resulted in a remarkably profitable thought for a wide category of structures: uniformly hyperbolic structures usually convey advanced evolution which, however, is now very well understood, either geometrically and statistically.

Another revolution has been happening within the final couple of many years, as one attempts to construct an international idea for

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In contrast, by a celebrated theorem of Herman [212], irrational rotations correspond to positive Lebesgue measure inside generic parametrized families. This kind of dichotomy is not uncommon in one-dimensional dynamics, as we shall see. 2. The Arnold family ha & : S1 —» S1 of circle maps is defined by [32] hajb(0) = O + a + bs]n2iTO modZ. 1) For |&| < 1/2TT, the map ha^ is a diffeomorphism. There is an open dense subset of values of a for which the rotation number is rational. However, by Herman's theorem, for a positive Lebesgue measure subset of values of a, the diffeomorphism ha^ is differentiably conjugate to an irrational rotation.

Quite in contrast, Hunt, Kaloshin [221] announce that the number of periodic points of period n is bounded by const exp(n 1+5 ), for a "full probability" subset of diffeomorphisms. 3 Hyperbolicity and fractal dimensions We are going to see that a refinement of the previous analysis leads to a deep connection between frequency of hyperbolicity in the unfolding of a homoclinic tangency and fractal dimensions of invariant sets. Let {ffj,)fj, be a smooth parametrized family of diffeomorphisms such that f — f0 has a quadratic homoclinic tangency q associated to a periodic point p, and this tangency is generically unfolded as the parameter \i varies: the stable manifold and the unstable manifold of the continuation p^ of p move with respect to each other with non-zero velocity near the point q, so that the tangency gives rise to a pair of transverse intersections when /i > 0.

1) Vi H2 Fig. 2. e. there are transverse intersections between their stable manifolds and unstable manifolds, and a tangency between Wu(p) and Ws(p2) may be created by another small perturbation of the diffeomorphism. 2. This is possible because the map