Unfolding chaotic quadratic maps --- parameter dependence of natural measures

We consider perturbations of quadratic maps $f_a$ admitting an absolutely continuous invariant probability measure, where $a$ is in a certain positive measure set $\mathcal{A}$ of parameters, and show that in any neighborhood of any such an $f_a$, we find a rich fauna of dynamics. There are maps with periodic attractors as well as non-periodic maps whose critical orbit is absorbed by the continuation of any prescribed hyperbolic repeller of $f_a$. In particular, Misiurewicz maps are dense in $\mathcal{A}$. Almost all maps $f_a$ in the quadratic family is known to possess a unique natural measure, that is, an invariant probability measure $\mu_a$ describing the asymptotic distribution of almost all orbits. We discuss weak*-(dis)continuity properties of the map $a\mapsto \mu_a$ near the set $\mathcal{A}$, and prove that almost all maps in $\mathcal{A}$ have the property that $\mu_a$ can be approximated with measures supported on periodic attractors of certain nearby maps. On the other hand, for any $a \in \mathcal{A}$ and any periodic repeller $\Gamma_a$ of $f_a$, the singular measure supported on $\Gamma_a$ can also approximated with measures supported on nearby periodic attractors. It follows that $a\mapsto \mu_a$ is not weak*continuous on any full-measure subset of $(0,2]$. Some of these results extend to unimodal families with critical point of higher order, and even to not-too-flat flat topped families

Complexities of differentiable dynamical systems

International audienceWe define the notion of localizable property for a dynamical system. Then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems. These notions are the fast growth of the number of periodic points, the positive entropy and the high emergence. We finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence