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.

We analyse the forced SIR model to investigate the influence of seasonality on the disease dynamics. Seasonality is often introduced in epidemic models via a transmission rate of sinusoidal type. In this talk, we present two results related to this model:

In 2017, Baraviera and Duarte extended a classical theorem from Le Page. They obtained an elegant proof for the local Holder continuity of the Lyapunov exponents of random linear cocycles defined over the Bernoulli Shift under generic hypothesis. The authors proved local Holder continuity with respect to the cocycle, with a fixed measure. The main tools are Furstenberg’s Formula and regularity properties from the stationary measure.

In this talk, we will present an estimate on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations. This convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we obtained similar results for affine skew product toral transformations and, in the case of one dimensional torus translation, these estimates are nearly optimal. This is a joint work with Xiao- Chuan Liu and Silvius Klein.