We study partially-hyperbolic skew-product maps over the Bernoulli shift with Hölder dependence on the base points. In the case of contracting fiber maps, symbolic blender-horseshoe is defined as an invariant set which meets any almost horizontal disk in a robust sense. These invariant sets are understood as blenders with center 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.1991 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.

We construct C 2 -robust homoclinic and heterodimensional tangencies of large codimension inside transitive partially hyperbolic sets.2010 Mathematics Subject Classification. 58F15, 58F17, 53C35.

Abstract. We study the dynamics of iterated function systems generated by a pair of circle diffeomorphisms close to rotations in the C 1+bv -topology. We characterize the obstruction to minimality and describe the limit set. In particular, there are no invariant minimal Cantor sets, which can be seen as a Denjoy/Duminy type theorem for iterated systems on the circle. Dédiéà G. Duminy

We show that C 1 -generically for diffeomorphisms of manifolds of dimension d ≥ 3, a homoclinic class containing saddles of different indices has a residual subset where the orbit of any point has historic behavior.

We show that any neighborhood of a non-degenerate reversible bifocal homoclinic orbit contains chaotic suspended invariant sets on N-symbols for all N ≥ 2. This will be achieved by showing switching associated with networks of secondary homoclinic orbits. We also prove the existence of super-homoclinic orbits (trajectories homoclinic to a network of homoclinic orbits), whose presence leads to a particularly rich structure. arXiv:1810.06359v2 [math.DS] 1 Jul 2019