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

摘要

We study the destruction of hyperbolic sets (horseshoes) in parametrized families of diffeomorphisms through homoclinic tangencies taking place inside the limit set. If the limit set at the tangency parameter has small dimension (limit capacity) then hyperbolicity prevails after the bifurcation (full Lebesgue density). We also prove that, if that limit set is thick then the system exhibits homoclinic tangencies for a whole parameter interval across the bifurcation. These results are based on a geometric analysis of the limit set at the tangency, including a statement of bounded distortion. [References: 13]