Skew product cycles with rich dynamics: From totally non-hyperbolic dynamics to fully prevalent hyperbolicity

Abstract

We introduce a two-parameter family of ‘partially hyperbolic’ skew products (Ga, t)a > 0, t ∈ [ − ε, ε] maps with one dimensional centre direction. In this family, the parameter a models the central dynamics and the parameter t the unfolding of cycles (that occurs for t = 0). The parameter a also measures the ‘central distortion’ of the systems: for small a, the distortion of the systems is small and it increases and goes to infinity as a → ∞. The family (Ga, t) displays some of the main characteristic properties of the unfolding of heterodimensional cycles as intermingled homoclinic classes of different indices and secondary bifurcations via collision of hyperbolic homoclinic classes. For a ∈ (0, log 2), the dynamics of (Ga, t) is always non-hyperbolic after the unfolding of the cycle. However, for a > log 4 intervals of t-parameters corresponding to hyperbolic dynamics appear and turn into totally prevalent as a → ∞ (the density of ‘hyperbolic parameters’ goes to 1 as a → ∞). The dynamics of the maps Ga, t is described using a family of iterated function systems modelling the dynamics in the one-dimensional central direction.