We study the minimal distance between two orbit segments of length $n$, in
a random dynamical system with sufficiently good mixing properties. For the annealed version of this problem, the asymptotic behavior is given by a dimension-like quantity associated to the invariant measure, called its correlation dimension. We study the analogous quenched question, and show that the asymptotic behavior is more involved: two correlation dimensions show up, giving rise to a non-smooth behavior of the associated asymptotic exponent.

We present a skew system made of two unimodal maps of the interval coupled in a master-slave configuration. This system arises in the physics literature as an explanatory model of the so-called topological synchronization, i.e. the mechanism believed to be at the basis of emerging collective phenomena in coupled chaotic systems.

This talk is divided in two parts.

Firstly, we analyze the effect of the seasonality in the SIR model, realized through the addition of a sinusoidal map in the transmission rate. Our major findings are the following:

Non-autonomous or random dynamical systems provide useful and flexible models to investigate systems whose evolution depends on external factors, such as noise and seasonal forcing. In the last fifteen years, transfer operators have been combined with multiplicative ergodic theory to shed light on ergodic-theoretic properties of random dynamical systems, through the so-called Lyapunov--Oseledets spectrum. While the scope of this framework is broad, in practice it is challenging to identify and even approximate this spectrum.

The expanding Lorenz map, which is obtained by taking a Poincaré section of the geometric Lorenz attractor and quotienting along stable leaves, has been studied in the literature.

We study the discrete centralizers of smooth flows and their time-t maps. Centralizers describe the symmetries of the respective dynamics. Given a $C^1$ -flow $(\varphi_t)_t$ on a compact manifold preserving an ergodic probability measure $\mu$, we prove that there exists a Baire generic subset of values of $t \in \mathbb{R}$ such that the centralizer of the time-$t$ map coincides with the discrete centralizer of the flow on $supp(\mu)$. The result relies on the ergodicity of generic time-t maps, established by Pugh and Shub in the seventies.

We consider nonuniformly hyperbolic dynamical systems with exponential decay of correlations, and show that exponential decay holds for pairs of observables v in L^p, w in L^q for which the corresponding local Holder constants also lie in L^p and L^q, with 1/p+1/q<1. Along the way, we give a particularly simple proof of quasicompactness for the one-sided transfer operator.

Title: About the stability of heteroclinic cycles: a new approach

Speaker: Alexandre Rodrigues (ISEG, CMUP) 

Abstract:

The study of random products of operators appears naturally in many
areas of mathematics and its applications. An example of this is product
of isometries of some metric space. Much like the Oseledets’ theorem
governs the behaviour of products of linear operators, the metric setting
can be described with the multiplicative ergodic theorem of Karlsson and
Gouezel [1].
In this talk we will focus on the specific case where the metric space
is a strongly hyperbolic. These spaces exhibit very nice properties with

 In this talk we discuss hyperbolicity, expansivity, the shadowing property and structural stability for operators on Banach spaces. In particular, we discuss relations between these notions and some open problems. All concepts will be gently introduced and the talk will be accessible to graduate students and non-specialists.