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

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 parameteramodels the central dynamics and the parametertthe unfolding of cycles (that occurs fort= 0). The parameteraalso measures the ‘central distortion’ of the systems: for smalla, the distortion of the systems is small and it increases and goes to infinity asa→ ∞. 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. Fora∈ (0, log 2), the dynamics of (Ga, t) is always non-hyperbolic after the unfolding of the cycle. However, fora> log 4 intervals oft-parameters corresponding to hyperbolic dynamics appear and turn into totally prevalent asa→ ∞ (the density of ‘hyperbolic parameters’ goes to 1 asa→ ∞). The dynamics of the mapsGa, tis described using a family of iterated function systems modelling the dynamics in the one-dimensional central direction. [ABSTRACT FROM AUTHOR]