Abstract
Our aim is to describe the “amount of nonhyperbolicity” either on an orbit-wise level or on a measure-wise one. This is done in the simplest setting of step skew products defined over a shift and with circle fiber maps, which provides a convenient model for partially hyperbolic diffeomorphisms with one-dimensional central bundle. Starting point is topologically transitive systems which display coexisting regions with different type of hyperbolicity. Transitivity forces an interaction between these regions and the emergence of nonhyperbolicity.We review three, closely related, main mechanisms to construct nonhyperbolic ergodic measures. One main approach is to ergodically approximate nonhyperbolicity by hyperbolicity.
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