Cours doctoraux

## Theorems and tools in one-dimensional dynamics

Type:
Site:
Date:
06/12/2017 - 13:00 - 15:00
Orateur:
DOBBS Neil
Localisation:
University College Dublin
Localisation:
Irlande
Résumé:

The first lecture will be an accessible introduction (peut-être en français) to smooth one-dimensional dynamics. After some historical examples and results, we shall consider the dynamics of some maps $f_a : x \mapsto ax(1-x)$ from the quadratic family.

In what follows, we shall investigate typical behaviour from the probabilistic viewpoint. In particular, we shall show, under certain conditions, the existence of probability measures which describe the statistical behaviour of Lebesgue almost every orbit. Time permitting, the Markov extension, together with its utility in proving results concerning continuity of statistical properties, will be presented.

## Theorems and tools in one-dimensional dynamics

Type:
Site:
Date:
04/12/2017 - 10:00 - 12:00
Salle:
P1 011
Orateur:
DOBBS Neil
Localisation:
University College Dublin
Localisation:
Irlande
Résumé:

The first lecture will be an accessible introduction (peut-être en français) to smooth one-dimensional dynamics. After some historical examples and results, we shall consider the dynamics of some maps $f_a : x \mapsto ax(1-x)$ from the quadratic family.

In what follows, we shall investigate typical behaviour from the probabilistic viewpoint. In particular, we shall show, under certain conditions, the existence of probability measures which describe the statistical behaviour of Lebesgue almost every orbit. Time permitting, the Markov extension, together with its utility in proving results concerning continuity of statistical properties, will be presented.

## Theorems and tools in one-dimensional dynamics

Type:
Site:
Date:
29/11/2017 - 14:00 - 16:00
Salle:
P1 011
Orateur:
DOBBS Neil
Localisation:
University College Dublin
Localisation:
Irlande
Résumé:

The first lecture will be an accessible introduction (peut-être en français) to smooth one-dimensional dynamics. After some historical examples and results, we shall consider the dynamics of some maps $f_a : x \mapsto ax(1-x)$ from the quadratic family.

In what follows, we shall investigate typical behaviour from the probabilistic viewpoint. In particular, we shall show, under certain conditions, the existence of probability measures which describe the statistical behaviour of Lebesgue almost every orbit. Time permitting, the Markov extension, together with its utility in proving results concerning continuity of statistical properties, will be presented.

## A variational approach linking solution of elliptic equations to the geometry of the domain

Type:
Site:
Date:
21/04/2017 - 10:00 - 12:00
Salle:
P1 038
Orateur:
MOLLE Riccardo
Localisation:
Université Rome 2
Localisation:
Italie

## A variational approach linking solution of elliptic equations to the geometry of the domain

Type:
Site:
Date:
20/04/2017 - 09:45 - 12:45
Salle:
P2 P43
Orateur:
MOLLE Riccardo
Localisation:
Université Rome 2
Localisation:
Italie

## A variational approach linking solution of elliptic equations to the geometry of the domain

Type:
Site:
Date:
19/04/2017 - 09:45 - 12:45
Salle:
P1 038
Orateur:
MOLLE Riccardo
Localisation:
Université Rome 2
Localisation:
Italie

## About thermodynamic formalism

Type:
Site:
Date:
25/11/2016 - 15:00 - 16:45
Salle:
P1 011
Orateur:
BRUIN Henk
Localisation:
Université de Vienne
Localisation:
Autriche
Résumé:

Thermodynamic formalism originated as an approach wihin statistical mechanics in physics, in order to explain, among other things, the changes of aggregation states (phase transitions). It was introduced into dynamical systems and mathematically formalised in the 1970s by the work of Sinai, Ruelle and Bowen, mainly in the context of hyperbolic (and symbolic) systems, and found several applications, including dimension theory and fractal geometry. Gradually, also non-uniformly hyperbolic systems entered the picture, with reappearance of phase transitions. The study thermodynamic properties of interval maps is part of this, and in these lectures I will discuss them in greater detail.

Among the topics covered:
- Historic context (including the Ising model)
- Hyperbolic dynamics and the Grifftih-Ruelle Theorem
- Non-uniformly hyperbolic dynamics: Intermittent maps vs Hofbauer potential
- Interval dynamics: Collet-Eckmann maps and Feigenbaum maps and everything in between.

The lectures will be on graduate level or up.

## About thermodynamic formalism

Type:
Site:
Date:
18/11/2016 - 15:00 - 16:45
Salle:
P1 011
Orateur:
BRUIN Henk
Localisation:
Université de Vienne
Localisation:
Autriche
Résumé:

Thermodynamic formalism originated as an approach wihin statistical mechanics in physics, in order to explain, among other things, the changes of aggregation states (phase transitions). It was introduced into dynamical systems and mathematically formalised in the 1970s by the work of Sinai, Ruelle and Bowen, mainly in the context of hyperbolic (and symbolic) systems, and found several applications, including dimension theory and fractal geometry. Gradually, also non-uniformly hyperbolic systems entered the picture, with reappearance of phase transitions. The study thermodynamic properties of interval maps is part of this, and in these lectures I will discuss them in greater detail.

Among the topics covered:
- Historic context (including the Ising model)
- Hyperbolic dynamics and the Grifftih-Ruelle Theorem
- Non-uniformly hyperbolic dynamics: Intermittent maps vs Hofbauer potential
- Interval dynamics: Collet-Eckmann maps and Feigenbaum maps and everything in between.

The lectures will be on graduate level or up.

## About thermodynamic formalism

Type:
Site:
Date:
15/11/2016 - 15:00 - 16:45
Salle:
P1 011
Orateur:
BRUIN Henk
Localisation:
Université de Vienne
Localisation:
Autriche
Résumé:

Thermodynamic formalism originated as an approach wihin statistical mechanics in physics, in order to explain, among other things, the changes of aggregation states (phase transitions). It was introduced into dynamical systems and mathematically formalised in the 1970s by the work of Sinai, Ruelle and Bowen, mainly in the context of hyperbolic (and symbolic) systems, and found several applications, including dimension theory and fractal geometry. Gradually, also non-uniformly hyperbolic systems entered the picture, with reappearance of phase transitions. The study thermodynamic properties of interval maps is part of this, and in these lectures I will discuss them in greater detail.

Among the topics covered:
- Historic context (including the Ising model)
- Hyperbolic dynamics and the Grifftih-Ruelle Theorem
- Non-uniformly hyperbolic dynamics: Intermittent maps vs Hofbauer potential
- Interval dynamics: Collet-Eckmann maps and Feigenbaum maps and everything in between.

The lectures will be on graduate level or up.

## Variational aspects of Liouville equations

Type:
Site:
Date:
02/06/2016 - 10:00 - 12:00
Salle:
201
Orateur:
MALCHIODI Andrea
Localisation:
SISSA Trieste
Localisation:
Italie
Résumé:

We consider a class of Liouville equations that arise in differential geometry when prescribing the Gaussian curvature of a surface and in models of mathematical physics describing stationary Euler flows and self-dual Chern-Simons equations. We discuss methods, variational in nature, to derive general existence results from suitable improvements of the Moser-Trudinger inequality combined with Morse-theoretical methods. We will treat in particular the case with Dirac masses representing, in the above motivations, conical singularities or vortex points.