Searcharxiv⌕ Search

arXiv subjects

Jorge Milhazes Freitas

Publications and source records attributed to Jorge Milhazes Freitas.

At least 19 recordsLinked to original sources
arXiv2026

Distributional results for the shortest distance between trajectories of different dynamics

We establish Extreme Value Distributions for the closest encounter between trajectories generated by different maps defined in the same reference phase space. For a class of strongly mixing maps, we show that the limit distribution depends on the length of the different trajectories and the co-dimension of the associated invariant measures. It is also modulated by an Extremal Index, that informs on the tendency of nearby points to diverge along with the evolution of their respective dynamics, serving as an indicator of their compatibility. We give a formula for this quantity for a class of chaotic maps of the interval and for the co-dimension in the case when the respective measures admit densities with isolated zeros and singularities. We present diverse examples of systems satisfying these assumptions and compute the different parameters modulating the limit distribution.

math.DS↗
arXiv2026

Multivariate extreme values for dynamical systems

We establish a theory for multivariate extreme value analysis of dynamical systems. Namely, we provide conditions adapted to the dynamical setting which enable the study of dependence between extreme values of the components of $\R^d$-valued observables evaluated along the orbits of the systems. We study this cross-sectional dependence, which results from the combination of a spatial and a temporal dependence structures. We give several illustrative applications, where concrete systems and dependence sources are introduced and analysed.

math.DS↗
arXiv2026

A functional limit theorem for a dynamical system with an observable maximised on a Cantor set

We consider heavy-tailed observables maximised on a dynamically defined Cantor set and prove convergence of the associated point processes as well as functional limit theorems. The Cantor structure, and its connection to the dynamics, causes clustering of large observations: this is captured in the `decorations' on our point processes and functional limits, an application of the theory developed in a paper by the latter three authors.

math.DS↗
arXiv2025

Enriched functional limit theorems for dynamical systems

We prove functional limit theorems for dynamical systems in the presence of clusters of large values which, when summed and suitably normalised, get collapsed in a jump of the limiting process observed at the same time point. To keep track of the clustering information, which gets lost in the usual Skorohod topologies in the space of càdlàg functions, we introduce a new space which generalises the already more general spaces introduced by Whitt. Our main applications are to hyperbolic and non-uniformly expanding dynamical systems with heavy-tailed observable functions maximised at dynamically linked maximal sets (such as periodic points). We also study limits of extremal processes and record times point processes for observables not necessarily heavy tailed. The applications studied include hyperbolic systems such as Anosov diffeomorphisms, but also non-uniformly expanding maps such as maps with critical points of Benedicks-Carleson type or indifferent fixed points such as Pomeau-Manneville or Liverani-Saussol-Vaienti maps. The main tool is a limit theorem for point processes with decorations derived from a bi-infinite sequence called the transformed anchored tail process.

math.DS↗
arXiv2024

Convergence to decorated Lévy processes in non-Skorohod topologies for dynamical systems

We present a general framework for weak convergence to decorated Lévy processes in enriched spaces of càdlàg functions for vector-valued processes arising in deterministic systems. Applications include uniformly expanding maps and unbounded observables as well as nonuniformly expanding/hyperbolic maps with bounded observables. The latter includes intermittent maps and dispersing billiards with flat cusps. In many of these examples, convergence fails in all of the Skorohod topologies. Moreover, the enriched space picks up details of excursions that are not recorded by Skorohod or Whitt topologies.

math.DS↗
arXiv2023

Inducing techniques for quantitative recurrence and applications to Misiurewicz maps and doubly intermittent maps

We prove an abstract result establishing that one can obtain the convergence of Rare Events Point Processes counting the number of orbital visits to a sequence of shrinking target sets from the convergence of corresponding point processes for some induced system and matching shadowing shrinking sets inside the base of the inducing scheme. We apply this result to prove a dichotomy for two classes of non-uniformly hyperbolic interval maps: Misiurewicz quadratic maps and doubly intermittent maps. The dichotomy holds in the sense that the shrinking target sets may accumulate in any individual point $ζ$ chosen in the phase space and then one either obtains a limiting homogeneous Poisson process at every non-periodic point $ζ$ or a limiting compound Poisson process with geometric multiplicity distribution at every periodic point. We also highlight the reconstruction performed in order to recover the multiplicity distribution for a periodic orbit sitting outside the base of the induced map.

math.DS↗
arXiv2019

Dynamical counterexamples regarding the Extremal Index and the mean of the limiting cluster size distribution

The Extremal Index is a parameter that measures the intensity of clustering of rare events and is usually equal to the reciprocal of the mean of the limiting cluster size distribution. We show how to build dynamically generated stochastic processes with an Extremal Index for which that equality does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised at at least two points of the phase space, where one of them is an indifferent periodic point and another one is either a repelling periodic point or a non periodic point. The occurrence of extreme events is then tied to the entrance and recurrence to the vicinities of those points. This enables to mix the behaviour of an Extremal Index equal to $0$ with that of an Extremal Index larger than $0$. Using bi-dimensional point processes we explain how mass escapes in order to destroy the usual relation. We also perform a study about the formulae to compute the limiting cluster size distribution introduced in \cite{FFT13,AFV15} and prove that ergodicity is enough to establish that the reciprocal of the Extremal Index is equal to the limit of the mean of the finite time cluster size distribution, which, in the case of the counterexamples given, does not coincide with the mean of the limit of the cluster size distribution.

math.DS↗
arXiv2019

Rare events for Cantor target sets

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.

math.DS↗
arXiv2018

Clustering indices and decay of correlations in non-Markovian models

When there is no independence, abnormal observations may have a tendency to appear in clusters instead of scattered along the time frame. Identifying clusters and estimating their size are important problems arising in statistics of extremes or in the study of quantitative recurrence for dynamical systems. In the classical literature, the Extremal Index appears associated to the cluster size and, in fact, it is usually interpreted as the reciprocal of the mean cluster size. This quantity involves a passage to the limit and in some special cases this interpretation fails due to an escape of mass when computing the limiting point processes. Smith \cite{S88} introduced a regenerative process exhibiting such disagreement. Very recently, in \cite{AFF18} the authors used a dynamical mechanism to emulate the same inadequacy of the usual interpretation of the Extremal Index. Here, we consider a general regenerative process that includes Smith's model and show that it is important to consider finite time quantities instead of asymptotic ones and compare their different behaviours in relation to the cluster size. We consider other indicators such as what we call the sojourn time, which corresponds to the size of groups of abnormal observations, when there is some uncertainty regarding where the cluster containing that group was actually initiated. We also study the decay of correlations of the non-Markovian models considered.

math.PR↗
arXiv2017

Complete convergence and records for dynamically generated stochastic processes

We consider empirical multi-dimensional Rare Events Point Processes that keep track both of the time occurrence of extremal observations and of their severity, for stochastic processes arising from a dynamical system, by evaluating a given potential along its orbits. This is done both in the absence and presence of clustering. A new formula for the piling of points on the vertical direction of bi-dimensional limiting point processes, in the presence of clustering, is given, which is then generalised for higher dimensions. The limiting multi-dimensional processes are computed for systems with sufficiently fast decay of correlations. The complete convergence results are used to study the effect of clustering on the convergence of extremal processes, record time and record values point processes. An example where the clustering prevents the convergence of the record times point process is given.

math.DS↗
arXiv2017

Extreme Value Laws for sequences of intermittent maps

We study non-stationary stochastic processes arising from sequential dynamical systems built on maps with a neutral fixed points and prove the existence of Extreme Value Laws for such processes. We use an approach developed in \cite{FFV16}, where we generalised the theory of extreme values for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. The present work is an extension of our previous results for concatenations of uniformly expanding maps obtained in \cite{FFV16}.

math.DS↗
arXiv2017

Convergence of Marked Point Processes of Excesses for Dynamical Systems

We consider stochastic processes arising from dynamical systems simply by evaluating an observable function along the orbits of the system and study marked point processes associated to extremal observations of such time series corresponding to exceedances of high thresholds. Each exceedance is marked by a quantity intended to measure the severity of the exceedance. In particular, we consider marked point processes measuring the aggregate damage by adding all the excesses over the threshold that mark each exceedance (AOT) or simply by adding the largest excesses in a cluster of exceedances (POT). We provide conditions to prove the convergence of such marked point processes to a compound Poisson process, for whose multiplicity distribution we give an explicit formula. These conditions are shown to follow from a strong form of decay of correlations of the system. Moreover, we prove that the convergence of the marked point processes for a `nice' first return induced map can be carried to the original system. The systems considered include non-uniformly expanding maps (in one or higher dimensions), maps with intermittent fixed points or non-recurrent critical points. For a general class of examples, the compound Poisson limit process is computed explicitly and, in particular, in the POT case we obtain a generalised Pareto multiplicity distribution.

math.DS↗
arXiv2017

Extreme Value Laws for non stationary processes generated by sequential and random dynamical systems

We develop and generalize the theory of extreme value for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. We apply our results to non-autonomous dynamical systems, in particular to {\em sequential dynamical systems}, given by uniformly expanding maps, and to a few classes of random dynamical systems. Some examples are presented and worked out in detail.

math.DS↗
arXiv2017

Rare Events for the Manneville-Pomeau map

We prove a dichotomy for Manneville-Pomeau maps $f:[0,1]\to [0, 1]$: given any point $ζ\in [0,1]$, either the Rare Events Point Processes (REPP), counting the number of exceedances, which correspond to entrances in balls around $ζ$, converge in distribution to a Poisson process; or the point $ζ$ is periodic and the REPP converge in distribution to a compound Poisson process. Our method is to use inducing techniques for all points except 0 and its preimages, extending a recent result by Haydn, Winterberg and Zweimüller, and then to deal with the remaining points separately. The preimages of 0 are dealt with applying recent results by Aytaç, Freitas and Vaienti. The point $ζ=0$ is studied separately because the tangency with the identity map at this point creates too much dependence, which causes severe clustering of exceedances. The Extremal Index, which measures the intensity of clustering, is equal to 0 at $ζ=0$, which ultimately leads to a degenerate limit distribution for the partial maxima of stochastic processes arising from the dynamics and for the usual normalising sequences. We prove that using adapted normalising sequences we can still obtain non-degenerate limit distributions at $ζ=0$.

math.DS↗
arXiv2016

Extreme Value Laws for dynamical systems with countable extremal sets

We consider stationary stochastic processes arising from dynamical systems by evaluating a given observable along the orbits of the system. We focus on the extremal behaviour of the process, which is related to the entrance in certain regions of the phase space, which correspond to neighbourhoods of the maximal set $\mathcal M$, i.e. the set of points where the observable is maximised. The main novelty here is the fact that we consider that the set $\mathcal M$ may have a countable number of points, which are associated by belonging to the orbit of a certain point, and may have accumulation points. In order to prove the existence of distributional limits and study the intensity of clustering, given by the Extremal Index, we generalise the conditions previously introduced in \cite{FFT12,FFT15}.

math.DS↗
arXiv2016

Extremes and Recurrence in Dynamical Systems

This book provides a comprehensive introduction for the study of extreme events in the context of dynamical systems. The introduction provides a broad overview of the interdisciplinary research area of extreme events, underlining its relevance for mathematics, natural sciences, engineering, and social sciences. After exploring the basics of the classical theory of extreme events, the book presents a careful examination of how a dynamical system can serve as a generator of stochastic processes, and explores in detail the relationship between the hitting and return time statistics of a dynamical system and the possibility of constructing extreme value laws for given observables. Explicit derivation of extreme value laws are then provided for selected dynamical systems. The book then discusses how extreme events can be used as probes for inferring fundamental dynamical and geometrical properties of a dynamical system and for providing a novel point of view in problems of physical and geophysical relevance. A final summary of the main results is then presented along with a discussion of open research questions. Finally, an appendix with software in Matlab programming language allows the readers to develop further understanding of the presented concepts.

math.DS↗