Symbolic blender-horseshoes and applications

Abstract

We study partially hyperbolic skew-product maps over the Bernoulli shift with Hölder dependence on the base points. In the case of contracting fibre maps, symbolic blender-horseshoe is defined as an invariant set which meets any almost horizontal disc in a robust sense. These invariant sets are understood as blenders with a centre stable bundle of any dimension. We then give necessary conditions (covering property) on an iterated function system such that the relevant skew-product has a symbolic blender-horseshoe. We use this local plug to yield robustly non-hyperbolic transitive diffeomorphisms and robust heterodimensional cycles of co-index equal to the dimension of the central direction.