I will discuss step skew-product maps with concave interval fiber maps over a certain subshift and analyze the structure of their spaces of ergodic measures. The fiber maps are assumed to have expanding and contracting regions. As a consequence, the skew-product has pairs of horseshoes of different type of hyperbolicity (fiber Lyapunov exponent) and also nonhyperbolic ergodic measures.

These 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. We study associated bifurcation phenomena that involve a jump of the space of ergodic measures and, in some cases, also of entropy.
(joint work with M. Rams and L.J. Díaz)