Speakers

Registration

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Registered Participants (by September 08)

Poster & Schedule

Program:

Elismar Oliveira

Title: 

Idempotent approach to level-2 variational principles in Thermodynamical Formalism

Abstract: 

In this joint work with A. O. Lopes and J. K. Mengue, we extend the idempotent formalism to the level-2 setting. We introduce the notions of idempotent pressure and an associated density entropy at level-2. The idempotent pressure emerges as a natural analogue of measure within the max-plus framework at this level.

In this general context, equilibrium states—those maximizing the variational principle—are not necessarily unique. We explore connections with the recently introduced general convex pressure for level-1 functions, developed by Bis, Carvalho, Mendes, and Varandas.

Our framework encompasses both dynamical and non-dynamical settings. Specifically, we study dynamical systems acting on the space of probability measures and construct level-2 idempotent pressure functions that remain invariant under the dynamics of the pushforward map, via a form of the Ruelle operator.

Yushi Nakano

Title: 

Metastable limit theorems in chaotic dynamical systems

Abstract: 

A metastable state is a state which is not a real stable state, but can be observed for a long time. This is a concept that appears in several areas of natural science, like chemical kinetics, meteorology, neuroscience, etc. Some of these phenomena can be modeled as a dynamical system, but traditional dynamical systems theory was developed by analyzing behaviors of the system in the infinite time limit, so mathematical theory for understanding (non-trivial) dynamics on metastable time scales would be far from complete (although there are several important developments recently). 

In this talk, I try to concentrate on a (famous) toy model dynamics, which is a piecewise expanding interval map without statistical stability (i.e. its "physical" invariant measure does not vary continuously under perturbations), to make the presentation of our idea/formulation transparent. Our results include strong laws of large numbers, central limit theorems with Berry-Esseen type error estimates, large deviation principles, almost sure invariant principles on metastable time scales. This is partially based on joint works in progress with J. Atnip,  C. Gonzalez-Tokman,  G. Froyland, and S. Vaienti.

Alexander Plakhov

Title: 

Extremal problems in billiards

Abstract: 

We consider the billiard in the exterior of a body — a compact set in R n (n ≥ 2) with piecewise smooth boundary. Within this model, we consider problems of minimal resistance in a particular direction and minimal resistance averaged over all directions. It turns out [1] that there are bodies with zero resistance, and also (using an optical analogy) bodies invisible in one direction. It is known [2] that bodies with zero resistance in all directions, and hence, perfectly (in all directions) invisible bodies do not exist. We consider the problem of least average resistance for a body of fixed volume contained in a unit sphere. This problem has not been completely solved. A lower bound for the average resistance, which is a function of body volume, is found [3]. This result is obtained using the methods of the vector-valued problem of optimal mass transport. 

References 

[1] A. Aleksenko and A. Plakhov. Bodies of zero resistance and bodies invisible in one direction. Nonlinearity 22, 1247-1258 (2009). 

[2] A. Plakhov and V. Roshchina. Invisibility in billiards. Nonlinearity 24, 847-854 (2011). 

[3] A. Plakhov and V. Roshchina. The problem of optimal camouflaging. SIAM J. Math. Anal. 57, 95-117 (2025).

Paulo Varandas

Title: 

Statistical properties of equilibrium states for linear cocycles

Abstract: 

Equilibrium states for hyperbolic maps and Holder continuous cocycles are known to exist and satisfy good statistical properties (including central limit theorems, almost sure invariance principle and exponential large deviations). Much less is known in case of equilibrium states for non-additive families of potentials.  I will present some recent results on the thermodynamic formalism of Holder continuous fiber-bunched matrix cocycles and discuss some applications to Anosov diffeomorphisms and hyperbolic repellers. This is based on a joint work with Reza Mohammadpour (Uppsala).

Room 11.1.32 of the Department of Mathematics, Universidade de Aveiro

How to arrive

From Lisbon:

There is a Metro leaving the airport and going to Estação do Oriente. There are regular trains from Lisbon to Aveiro, just pay attention to different duration of the trip and prices.

From Porto Airport:

There is a Flixbus direct bus from Porto Airport to the train station in Aveiro.  Alternatively, there is a Metro leaving the airport and going to the Campanhã Train Station. There are regular trains from Porto to Aveiro, just pay attention to different duration of the trip and prices.

In Aveiro:

The Mathematics Department of the University of Aveiro is located at 2.4 km from the Train Station, and it takes about 30 minutes walking. There is also the option of taking a Taxi or an Uber.