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Introduction to the Modern Theory of Dynamical Systems

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An Introduction To The Theory Of Point Processes

The Bootstrap and Edgeworth Expansion

An Introduction to Probability Theory and its Applications, Volume I

We review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deterministic modeling of extreme events, via continuous and discrete dynamic models The applications include climatic, seismic and socio-economic events, along with their prediction Two important results refer to (i) the complementarity of spectral analysis of a time series in terms of the continuous and the discrete part of its power spectrum; and (ii) the need for coupled modeling of natural and socio-economic systems Both these results have implications for the study and prediction of natural hazards and their human impacts

https://npg.copernicus.org/articles/18/295/2011/npg-18-295-2011.pdf

Extreme events: dynamics, statistics and prediction

We consider families of multimodal interval maps with polynomial growth of the derivative along the critical orbits. For these maps Bruin and Todd have shown the existence and uniqueness of equilibrium states for the potential $\phi_t:x\mapsto-t\log|Df(x)|$, for $t$ close to 1. We show that these equilibrium states vary continuously in the weak$^*$ topology within such families. Moreover, in the case $t=1$, when the equilibrium states are absolutely continuous with respect to Lebesgue, we show that the densities vary continuously within these families.

Statistical stability of equilibrium states for interval maps

We present conditions on families of diffeomorphisms that guarantee statistical stability and SRB entropy continuity. They rely on the existence of horseshoe-like sets with infinitely many branches and variable return times. As an application we consider the family of Henon maps within the set of Benedicks-Carleson parameters.

Statistical stability and continuity of SRB entropy for systems with Gibbs-Markov structures

We consider the extreme value theory of a hyperbolic toral automorphism $T: \mathbb{T}^2 \to \mathbb{T}^2$ showing that if a H\"older observation $\phi$ which is a function of a Euclidean-type distance to a non-periodic point $\zeta$ is strictly maximized at $\zeta$ then the corresponding time series $\{\phi\circ T^i\}$ exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $\phi$ and with extremal index one. If however $\phi$ is strictly maximized at a periodic point $q$ then the corresponding time-series exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $\phi$ but with extremal index not equal to one. We give a formula for the extremal index (which depends upon the metric used and the period of $q$). These results imply that return times are Poisson to small balls centered at non-periodic points and compound Poisson for small balls centered at periodic points.

/pdf/extremal-dichotomy-for-uniformly-hyperbolic-systems-234byj4xlk.pdf

Extremal dichotomy for uniformly hyperbolic systems

We study the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to a Cantor set of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index, which turns out to be very useful to identify the compatibility between the dynamics and the fractal structure of the limiting Cantor set. The computation of the Extremal Index is connected to the box dimension of the intersection between the Cantor set and its iterates.

Jorge Milhazes Freitas

Papers

Statistical stability of equilibrium states for interval maps

Statistical stability and continuity of SRB entropy for systems with Gibbs-Markov structures

Extremal dichotomy for uniformly hyperbolic systems

Rare events for Cantor target sets

A fast and environment-friendly MEPSPEP/UHPLC-PDA methodology to assess 3-hydroxy-4,5-dimethyl-2(5H)-furanone in fortified wines.