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 prove analytically the existence of chaotic dynamics in the forced SIR model. Although numerical experiments have already suggested that this model can exhibit chaotic dynamics, a rigorous proof (without computer-aided) was not given before. Under seasonality in the transmission rate, the coexistence of low birth and mortality rates with high recovery and transmission rates produces infinitely many periodic and aperiodic patterns together with sensitive dependence on the initial conditions.

Abstract. Every quasi-attractor of an iterated function system (IFS) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. As a consequence, we obtain that an IFS of homeomorphisms of the circle is renderable by the deterministic chaos game if and only if it is forward and backward minimal. This result provides examples of attractors (a forward but no backward minimal IFS on the circle) that are not renderable by the deterministic chaos game. We also prove that every well-fibred quasi-attractor is renderable by the deterministic chaos game as well as quasi-attractors of both, symmetric and non-expansive IFSs.