Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies (Q2722665)

Consider families of diffeomorphisms depending on a parameter \(c\), and generating horseshoes. Therefore a limit set, closure of the union of the backward and forward accumulation sets of all the orbits, exists. This paper studies the destruction of hyperbolic sets (horseshoes), when homoclinic tangencies bifurcations take place inside the limit set. This means that the situation corresponds to a first tangency (for \(c=0\)), no tangency occurring for \(c>0\). The text introduces an open set \(A\) of families unfolding such a homoclinic tangency, and recalls the definition of the ''limit capacity'' fractal dimension of a compact set. The author proves the following results.NEWLINENEWLINENEWLINE(a) For each family of the above open set A, the limit set is hyperbolic for \(c>0\).NEWLINENEWLINENEWLINE(b) If for \(c=0\) the limit set has a limit capacity smaller than 1, then this parameter value is a point of full Lebesgue density for the parameter set for which the limit set is hyperbolic.NEWLINENEWLINENEWLINE(c) A non-empty set inside \(A\) exists such that, for any family of this subset, the limit set for \(c=0\) has a limit capacity smaller than 1.NEWLINENEWLINENEWLINE(d) Let \(I\) be a parameter interval of negative values of \(c\). A non-empty set inside \(A\) exists such that, for any family of this subset, the diffeomorphism has some homoclinic tangency for a dense subset of the interval \(I\). In other words, if at the tangency the limit set has a small dimension, then the hyperbolicity prevails after the bifurcation (full Lebesgue density).NEWLINENEWLINENEWLINEIf the limit set is thick, the diffeomorphisms present homoclinic tangencies for a whole parameter interval across the bifurcation. The proof is based on a geometric analysis of the limit set at the tangency, including a statement of bounded distortion.NEWLINENEWLINENEWLINEIt would be desirable that the author places her results with respect to the relatively recent ones of V. S. Gonchenko, D. V. Turaev and L. P. Shil'nikov, not quoted in the references and in the historical part of the introduction. A tutorial presentation of this Russian contribution is given by \textit{L. Shil'nikov} [Mathematical problem of nonlinear dynamics: a tutorial. Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 1953-2001 (1997; Zbl 0909.58008), see also World Sci. Ser. Nonlinear Sci., Ser. A 26, 69-156 (1999; Zbl 0973.37001)].