Heterodimensionality of skew-products with concave fiber maps

Abstract: I will present some examples of skew-products with concave interval fiber maps over a certain subshift. Here the subshift occurs as the projection of those orbits that stay in a given neighborhood and gives rise to a new type of symbolic space which is (essentially) coded. The fiber maps have expanding and contracting regions. As a consequence, the skew-product dynamics has pairs of horseshoes of different type of hyperbolicity. In some cases, they dynamically interact due to the superimposed effects of the (fiber) contraction and expansion, leading to nonhyperbolic dynamics that is reflected on the ergodic level (existence of nonhyperbolic ergodic measures). The space of ergodic measures of the shift space is shown to be an entropy-dense Poulsen simplex, ergodic measures lift canonically to ergodic measures for the skew-product. Such skew-products can be embedded in increasing entropy one-parameter family of diffeomorphisms which stretch from a heterodimensional cycle to a collision of homoclinic classes. I will discuss some ingredients of associated bifurcation phenomena that involve a jump of the space of ergodic measures and, in some cases, also of entropy. (Joint work with L.J.Díaz and M.Rams)

dynamical systems

Audience: researchers in the topic

Dynamical systems seminar at the Jagiellonian University