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ByCeMEAI

Mini course – Topics in Random Dynamical Systems

The mini-course covers:

Breiman Ergodic Theorem, which states that for any Markov chain on a compact metric space with a continuous transition probability, the empirical measures accumulate onto stationary measures. Its proof will use transfer operator (acting on L^1 observables), Markov operator (acting on probability measures), and a martingale argument due to Furstenberg.

Invariant measures for skew-products with one-sided shift in the base vs. with two-sided shift in the base. Its proof will use “disintegration” of a measure, and another martingale argument.

U-state measures, and their correspondence with stationary measures.

Potentially, more results on “synchronization”, from the theoretical Random Dynamical Systems point of view.

This mini-course shall be very accessible to anyone who is proficient in Real Analysis (and some basic Measure Theory). In particular, you do not need to know everything or anything listed above; they are there just in case some key words might motivate you more than the mere title. Also, the Probability Theory tools will be supplied as needed.

Videos
Slides
References
ByEdmilson Santos

Impulsive Dynamical Systems

Everaldo de Mello Bonotto

Abstract:

In this talk, we present the concept of impulsive dynamical systems. Moreover, we establish sufficient conditions to obtain the existence of a global attractor.

ByEdmilson Santos

Lyapunov exponents, the problem of regularity

El Hadji Yaya Tall

Abstract:

The study of Lyapunov exponent goes back to the stability theory for differential equations developed in the doctoral thesis of Aleksandr M. Lyapunov, in the late 19th century, since then, it grew into a very broad area and active field in ergodic theory and dynamical systems, with several outstanding problems and applications.  In the early 80’s, Ricardo Mañé observed that the Lyapunov exponents of continuous 2-dimensional cocycle can be cancelled by arbitrarily small perturbation of the cocycle. The proof of this observation was completed by Jairo Bochi.

In 2010, Carlos Bocker and Marcelo Viana have proven that the Lyapunov exponent of random 2-dimensional cocycle vary always continuously with respect to the probability distribution. This result was also extended in arbitrarily dimensional by Artur Avila, Alex Eskin and Marcelo Viana. Recently Marcelo Viana and myself have proven that the Lyapunov exponents of random 2-dimensional cocycle are Hölder continuous functions of the underlying probability distribution at each point with simple Lyapunov spectrum. Moreover, they are log-Hölder continuous at every point.

I will first discuss the problem of continuity and give some ideas and results about Hölder continuity.

ByEdmilson Santos

Foliated Brownian motion and Harmonic measures.

Diego Sebastian Ledesma

Abstract:

For a foliated manifold, we present the construction of foliated Brownian motion via stochastic calculus adapted to foliation. The stochastic approach together with a proposed foliated vector calculus provide a natural method to work on harmonic measures. Also, we show a decomposition of the Laplacian in terms of the foliated and basic Laplacians, a characterization of totally invariant measures and a differential equation for the density of harmonic measures.

ByEdmilson Santos

Contribuições ao problema da continuidade para expoentes de Lyapunov

Adriana Sánchez Chavarría

Abstract:

Varios autores tem estudado o problema da continuidade dos expoentes de Lyapunov. Exemplo disto são os trabalhos de Bochi-Mañé, Bocker-Viana e Backes, Butler e Brown. No primeiro, os autores mostraram que todo SL(2)-cociclo contínuo que não é uniformemente hiperbólico pode ser aproximado por outro com expoentes nulos. Os outros dois mostram continuidade dos expoentes para o produto aleatorio de matrices com medidas de soporte compacto e quando a base é um deslocamento de tipo finito respetivamente.

Nesta palestra explicarei estos resultados e as principais diferenças respeto a os casos parcialmente hiperbólico e medidas com soporte não compacto.

ByEdmilson Santos

Geometric Anosov Actions

Uirá Norberto Matos de Almeida

Abstract:

There exists a long standing conjecture about the algebricity of Anosov actions of higher ranked abelian groups. On this talk, we present some work in progress on a especial cases of this conjecture, based on a similar work about the algebricity of Anosov Flows on dimension 5, by Young Fang. We consider an Anosov action with smooth invariant stable and unstable bundles that preserves a pseudo-metric, and present some partial results on the algebricity of such actions.

ByEdmilson Santos

Onset of diffusion in the kicked Harper model

Fabio Tal

Abstract:

We study a standard two-parameter family of area-preserving torus diffeomorphisms, known as the kicked Harper model in theoretical physics, by a combination of topological arguments and KAM-theory. We concentrate on the structure of the parameter sets where the rotation set has empty and non-empty interior, respectively, and describe their qualitative properties and scaling behaviour both for small and large parameters. This confirms numerical observations about the onset of diffusion in the physics literature. As a byproduct, we obtain the continuity of the rotation set within the class of Hamiltonian torus homeomorphisms.
This is Joint work w. T. Jäger and A. Koropecki

ByEdmilson Santos

Lift problem and equilibrium states

Vilton Pinheiro

Abstract:

In the first lecture we will talk about the problem of lifting a measure to an induced map. We will give a necessary and sufficient condition for a measure to be liftable as well as a condition for the lift to be ergodic and unique.

The second lecture will be dedicated to construct induced maps well adapted to a given ergodic invariant probability with all its Lyapunov exponents being positive (expanding measure).

Finally, in the third lecture, we will discuss about equilibrium states on the support of an ergodic invariant expanding probability. In particular, (1)  we will show that Viana maps have one and only one probability maximizing their entropy and (2) we will analyze the existence and uniqueness of the equilibrium states for Hölder potentials at high temperature.

ByEdmilson Santos

Renormalization of multicritical circle maps

Gabriela Estevez

Abstract:

In this talk, we will present the renormalization operator for multicritical circle maps and we will give certain conditions to get rigidity results, for almost all irrational rotation number. This is a joint work with Pablo Guarino (UFF-Brazil).

ByEdmilson Santos

Hamiltonian stability on open surfaces

Roland Rabanal

Abstract:

We extend to open surfaces with simple singularities the theorem proposed by Xavier Jarque and Zbigniew Nitecki for Hamiltonian flows in the plane that are structurally stable among Hamiltonian flows. We describe the Hamiltonian dynamics on open surfaces by presenting characterization theorems for Hamiltonian stability and some natural consequences.

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