Statistical properties and rare events for chaotic dynamical systems

Abstract

The application of Extreme Value Theory to Dynamical Systems has been a topic of interest for a few years now (see the influential work in [FFT10] which built on [Col01] and [FF08]) opening up a whole new framework for the statistical study of chaotic systems. In the early stages, the relationship between orbit visits to small sets of the ambient space and extreme values (of suitable random variables) provided statistical laws for recurrence (eg. [FFT10],[FFT11],[FFT12],[CFF⁺15]). In particular, strong recurrence properties such as periodicity directly impact on the limiting law due to them being responsible for the clustering of extreme observations. This study extends the scope of the classical Poincaré Recurrence Theorem in ergodic theory.

Recently, with the focus on functional limit theorems, very strong results for the convergence of (appropriately scaled) sums of heavy-tailed dynamically defined random variables have been deduced ([FFT20]). The most well-known distributional result for an appropriately scaled sum is the Central Limit Theorem which can’t be used with heavy tails.

The main purpose of our work is the application of the enriched functional limit theorem for heavy-tailed dynamical sums proved in [FFT20] (Theorem 2.2.6) to two different contexts which have previously been investigated from the point of view of extreme value laws: correlated maximal sets ([AFFR16] and [AFFR17]) and a Cantor maximal set ([FFRS20]). That essentially demands the convergence of some point processes, the key being the understanding of the clustering patterns of the tail observations of such processes. These patterns are well described by means of a structure introduced in [FFT20] and tailored to the dynamical context, which we prove to be, in the correlated maxima setting, as in Theorem 3.1.2, Theorem 3.1.14 or Theorem 3.2.1, and in the Cantor setting as in Theorem 4.5.1. Prior to our work, only a maximal set consisting of a single repelling periodic point had been considered. As we will see, the clustering patterns that we capture in our study are significantly richer (than for a maximal set reduced to a single point) and more accurately described (compared to the framework available for [AFFR16], [AFFR17] and [FFRS20]) via the new tool kit at hand.

We structure this thesis as follows. In Chapter 2, we summarise the background theory from [FFT20] which is required to the statement of the main Theorem 2.2.6 as well as to providing some insight into it. Then, we justify that the theorem can be deduced in the setting of correlated maximal sets, in Chapter 3, and in the setting of a Cantor maximal set, in Chapter 4.