Hale Aytaç - Mini-course

Mini-curso: Uma breve introdução à Teoria dos Valores Extremos

Teoria dos Valores Extremos (TVE) é um ramo da probabilidade que estuda o comportamento extremal de um processo estocástico. Nesse mini-curso, daremos uma breve introdução à TVE no contexto de sistemas dinâmicos, i. e., o processo estocástico tem sua origem em um sistema dinâmico caótico. Introduziremos também a Estatística do Tempo de Entrada/Retorno (ETE/ETR) em conjuntos arbitrariamente pequenos. TVE e ETE/ETR são dois lados da mesma moeda no estudo dos eventos raros, i.e., eventos que possuem baixa probabilidade de ocorrência. Mostraremos a relação entre esses dois conceitos.

Sandro Gallo - Conference

Return times, multi-fractal spectrum and large deviation

Behind this somewhat pedantic title, something very simple: the computation of the cumulant (free energy) of the return times. We will explore two main consequence of this result, (1) implications concerning the "dimension'' of the measure, and (2) an explicit large deviation principle for the return time statistics. This is a work in progress in collaboration with Jean-René Chazottes and Miguel Abadi.

Adriana Coutinho - Conference

LARGE DEVIATION FOR RETURN TIMES

Abstract. We prove a large deviation result for return times of the orbits of a dynamical system in a r-neighbourhood of an initial point x. Our result may be seen as a differentiable version of the work by Jain and Bansal who considered the return time of a stationary and ergodic process defined in a space of infinite sequences. This is joint work with J. Rousseau and B. Saussol.

Miguel Abadi - Conference

Kac, clustering and the extremal index

In this joint work with A. c. Freitas and J. Freitas we present how a motification and reinterpretation of Kac’s lemma

can be used to related the clustering phenomena to the extremal index. We treat the finite and infinite case. The clustering for single or multi periodic events are considered, which can be used to describe elastic clustering.

Isaia Nisoli - Conference

Algumas aplicações da aproximação rigorosa de medidas invariantes

Nessa palestra vou apresentar as técnicas de aproximação rigorosa de medidas invariantes e discutir algumas das aplicações delas: - estimativas rigorosas da entropia métrica em mapas expansoras unidimensionais - prova assistida por computador do Noise Induced Order no modelo de Matsumoto-Tsuda da reação de Belozouv-Zhabotinsky Nos três casos, o principio fundamental que nos permite aproximar a medida invariante são as propriedades de regolarização do operador de transferência em espaços de Banach específicos. Essas propriedades regularizadoras nos permitem de approximar os sistema dinâmicos com cadeias de Markov com um numero finito de estados, cujo comportamento aproxima o comportamento do sistema (estabilidade espetral).

Alejandra Rada - Conference

Ornstein and Weiss theorem for entrance time and Rényi Entropy

For ergodic systems with generating partitions, the well known result of Ornstein and Weiss shows that the exponential growth rate of the recurrence time is almost surely equal to the metric entropy. Here we look at the exponential growth rate of entrance times, and show that it equals the entropy, where the convergence is in probability in the product measure. This is however under the assumptions that the limiting entrance times distribution exists almost surely. This condition looks natural in the light of an example by Shields in which the limsup in the exponential growth rate is infinite almost everywhere but where the limiting entrance times do not exist. We then also consider phi-mixing systems and prove a result connecting the Rényi entropy to sums over the entrance times orbit segments. Joint with N. Haydn and M. Ko and C. Gupta.

Davide Azevedo - Conference

Clustering of extreme events created by multiple correlated maxima

We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. We will consider observables that achieve a global maximum value at multiple points, all belonging to the orbit of the same point, which may be periodic or not. We will see what impact this has on the Extremal Index and clustering patterns when compared to the case where the maximum is achieved in a single point. In particular we will observe the appearance of clustering not caused by periodic orbits. Joint work with A. C. M. Freitas, J. M. Freitas and F.B. Rodrigues.

Samuel Senti - Conference

Fluctuations of Ergodic Sums on Periodic Orbits under Specification.

In this joint work with Manferd Denker (Penn State) and Xuan Zhang (IME-USP), we study the fluctuations of ergodic sums by the means of global and local specifications on periodic points and obtain Lindeberg-type central limit theorems. As an application we can prove the weak convergence of ergodic sums to a mixture of normal distributions for systems with a unique measure of maximal entropy. Our results suggest to decompose the variances of ergodic sums according to global and local sources.