Jorge Milhazes Freitas

The extremal index, hitting time statistics and periodicity

Abstract The extremal index appears as a parameter in Extreme Value Laws for stochastic processes, characterising the clustering of extreme events. We apply this idea in a dynamical systems context to analyse the possible Extreme Value Laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures. We derive new, easily checkable, conditions which identify Extreme Value Laws with particular extremal indices. In the dynamical context we prove that the extremal index is associated with periodic behaviour. The analogy of these laws in the context of hitting time statistics, as studied in the authors’ previous works on this topic, is explained and exploited extensively allowing us to prove, for the first time, the existence of hitting time statistics for balls around periodic points. Moreover, for very well behaved systems (uniformly expanding) we completely characterise the extremal behaviour by proving that either we have an extremal index less than 1 at periodic points or equal to 1 at any other point. This theory then also applies directly to general stochastic processes, adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests.

Laws of Rare Events for Deterministic and Random Dynamical Systems

The object of this paper is twofold. From one side we study the dichotomy, in terms of the Extremal Index of the possible Extreme Value Laws, when the rare events are centred around periodic or non periodic points. Then we build a general theory of Extreme Value Laws for randomly perturbed dynamical systems. We also address, in both situations, the convergence of Rare Events Point Processes. Decay of correlations against L 1 observables will play a central role in our investigations.

Extreme values for Benedicks–Carleson quadratic maps

We consider the quadratic family of maps given by

The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics

f_{a}(x)=1-a x^2

Extremes and Recurrence in Dynamical Systems

with

Continuity of SRB measure and entropy for Benedicks–Carleson quadratic maps

x\in [-1,1]

Extreme Value Statistics for Dynamical Systems with Noise

, where

Extremal behaviour of chaotic dynamics

a

Speed of convergence for laws of rare events and escape rates

is a Benedicks-Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes

Convergence of rare event point processes to the Poisson process for planar billiards

X_0,X_1,...