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Mark Holland

Publications and source records attributed to Mark Holland.

16 recordsLinked to original sources
arXiv2024

Limitations of the Generalized Pareto Distribution-based estimators for the local dimension

Two dynamical indicators, the local dimension and the extremal index, used to quantify persistence in phase space have been developed and applied to different data across various disciplines. These are computed using the asymptotic limit of exceedances over a threshold, which turns to be a Generalized Pareto Distribution in many cases. However the derivation of the asymptotic distribution requires mathematical properties which are not present even in highly idealized dynamical systems, and unlikely to be present in real data. Here we examine in detail issues that arise when estimating these quantities for some known dynamical systems with a particular focus on how the geometry of an invariant set can affect the regularly varying properties of the invariant measure. We demonstrate that singular measures supported on sets of non-integer dimension are typically not regularly varying and that the absence of regular variation makes the estimates resolution dependent. We show as well that the most common extremal index estimation method is ambiguous for continuous time processes sampled at fixed time steps, which is an underlying assumption in its application to data.

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arXiv2024

On distributional limit laws for recurrence

For a probability measure preserving dynamical system $(\mathcal{X},f,\mu)$, the Poincar\'e Recurrence Theorem asserts that $\mu$-almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process $X_n(x)=\text{dist}(f^n(x),x))$, and real-valued functions thereof. For a wide class of non-uniformly expanding dynamical systems, we show that the time-$n$ counting process $R_n(x)$ associated to the number recurrences below a certain radii sequence $r_n(\tau)$ follows an \emph{averaged} Poisson distribution $G(\tau)$. Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process $X_n$.

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arXiv2023

Runs of Extremes of Observables on Dynamical Systems and Applications

We use extreme value theory to estimate the probability of successive exceedances of a threshold value of a time-series of an observable on several classes of chaotic dynamical systems. The observables have either a Fréchet (fat-tailed) or Weibull (bounded) distribution. The motivation for this work was to give estimates of the probabilities of sustained periods of weather anomalies such as heat-waves, cold spells or prolonged periods of rainfall in climate models. Our predictions are borne out by numerical simulations and also analysis of rainfall and temperature data.

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arXiv2021

Dichotomy results for eventually always hitting time statistics and almost sure growth of extremes

Suppose $(f,\mathcal{X},μ)$ is a measure preserving dynamical system and $ϕ\colon \mathcal{X} \to \mathbb{R}$ a measurable function. Consider the maximum process $M_n:=\max\{X_1 \ldots,X_n\}$, where $X_i=ϕ\circ f^{i-1}$ is a time series of observations on the system. Suppose that $(u_n)$ is a non-decreasing sequence of real numbers, such that $μ(X_1>u_n)\to 0$. For certain dynamical systems, we obtain a zero--one measure dichotomy for $μ(M_n\leq u_n\,\textrm{i.o.})$ depending on the sequence $u_n$. Specific examples are piecewise expanding interval maps including the Gauss map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences $u_n$. Our results on the permitted sequences $u_n$ are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory.

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arXiv2020

Limit theorems for wobbly interval intermittent maps

We consider perturbations of interval maps with indifferent fixed points, which we refer to as wobbly interval intermittent maps, for which stable laws for general Hölder observables fail. We obtain limit laws for such maps and Hölder observables. These limit laws are similar to the classical semistable laws previously established for random processes, but certain limitations imposed by the current dynamical set up are reflected in the main result. One of the considered examples is an interval map with a countable number of discontinuities, and to analyse it we need to construct a Markov/Young tower.

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arXiv2020

Extremes and extremal indices for level set observables on hyperbolic systems

Consider an ergodic measure preserving dynamical system $(T,X,μ)$, and an observable $ϕ:X\to\mathbb{R}$. For the time series $X_n(x)=ϕ(T^{n}(x))$, we establish limit laws for the maximum process $M_n=\max_{k\leq n}X_k$ in the case where $ϕ$ is an observable maximized on a curve or submanifold, and $(T,X,μ)$ is a hyperbolic dynamical system. Such observables arise naturally in weather and climate applications. We consider the extreme value laws and extremal indices for these observables on Anosov diffeomorphisms, Sinai dispersing billiards and coupled expanding maps. In particular we obtain clustering and nontrivial extremal indices due to self intersection of submanifolds under iteration by the dynamics, not arising from any periodicity.

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arXiv2019

Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables

We establish quantitative results for the statistical be\-ha\-vi\-our of \emph{infinite systems}. We consider two kinds of infinite system: i) a conservative dynamical system $(f,X,μ)$ preserving a $σ$-finite measure $μ$ such that $μ(X)=\infty$; ii) the case where $μ$ is a probability measure but we consider the statistical behaviour of an observable $ϕ\colon X\to[0,\infty)$ which is non-integrable: $\int ϕ\, dμ=\infty$. In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly oscillating. However, if the system has superpolynomial decay of correlations or has a Markov structure, then we show this oscillation cannot happen. In this case we prove asymptotic relations between the behaviour of $ϕ$, the local dimension of $μ$, and on the growth of Birkhoff sums (as time tends to infinity). We then establish several important consequences which apply to infinite systems of the kind i). This includes showing anomalous scalings in extreme event limit laws, or entrance time statistics. We apply our findings to non-uniformly hyperbolic systems preserving an infinite measure, establishing anomalous scalings in the case of logarithm laws of entrance times, dynamical Borel--Cantelli lemmas, almost sure growth rates of extremes, and dynamical run length functions.

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arXiv2017

Weak convergence to extremal processes and record events for non-uniformly hyperbolic dynamical systems

For a measure preserving dynamical system $(\mathcal{X},f, μ)$, we consider the time series of maxima $M_n=\max\{X_1,\ldots,X_n\}$ associated to the process $X_n=ϕ(f^{n-1}(x))$ generated by the dynamical system for some observable $ϕ:\mathcal{X}\to\mathbb{R}$. Using a point process approach we establish weak convergence of the process $Y_n(t)=a_n(M_{[nt]}-b_n)$ to an extremal process $Y(t)$ for suitable scaling constants $a_n,b_n\in\mathbb{R}$. Convergence here taking place in the Skorokhod space $\mathbb{D}(0,\infty)$ with the $J_1$ topology. We also establish distributional results for the record times and record values of the corresponding maxima process.

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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.

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arXiv2015

On the mixing properties of piecewise expanding maps under composition with permutations

We consider the effect on the mixing properties of a piecewise smooth interval map $f$ when its domain is divided into $N$ equal subintervals and $f$ is composed with a permutation of these. The case of the stretch-and-fold map $f(x)=mx \bmod 1$ for integers $m \geq 2$ is examined in detail. We give a combinatorial description of those permutations $σ$ for which $σ\circ f$ is still (topologically) mixing, and show that the proportion of such permutations tends to $1$ as $N \to \infty$. We then investigate the mixing rate of $σ\circ f$ (as measured by the modulus of the second largest eigenvalue of the transfer operator). In contrast to the situation for continuous time diffusive systems, we show that composition with a permutation cannot improve the mixing rate of $f$, but typically makes it worse. Under some mild assumptions on $m$ and $N$, we obtain a precise value for the worst mixing rate as $σ$ ranges through all permutations; this can be made arbitrarily close to $1$ as $N \to \infty$ (with $m$ fixed). We illustrate the geometric distribution of the second largest eigenvalues in the complex plane for small $m$ and $N$, and propose a conjecture concerning their location in general. Finally, we give examples of other interval maps $f$ for which composition with permutations produces different behaviour than that obtained from the stretch-and-fold map.

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arXiv2015

Quantitative recurrence statistics and convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems

For non-uniformly hyperbolic dynamical systems we consider the time series of maxima along typical orbits. Using ideas based upon quantitative recurrence time statistics we prove convergence of the maxima (under suitable normalization) to an extreme value distribution, and obtain estimates on the rate of convergence. We show that our results are applicable to a range of examples, and include new results for Lorenz maps, certain partially hyperbolic systems, and non-uniformly expanding systems with sub-exponential decay of correlations. For applications where analytic results are not readily available we show how to estimate the rate of convergence to an extreme value distribution based upon numerical information of the quantitative recurrence statistics. We envisage that such information will lead to more efficient statistical parameter estimation schemes based upon the block-maxima method.

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arXiv2015

Extremal dichotomy for uniformly hyperbolic systems

We consider the extreme value theory of a hyperbolic toral automorphism $T: \mathbb{T}^2 \to \mathbb{T}^2$ showing that if a Hölder observation $ϕ$ which is a function of a Euclidean-type distance to a non-periodic point $ζ$ is strictly maximized at $ζ$ then the corresponding time series $\{ϕ\circ T^i\}$ exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $ϕ$ and with extremal index one. If however $ϕ$ 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 $ϕ$ 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.

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arXiv2012

Dimension results for inhomogeneous Moran set constructions

We compute the Hausdorff, upper box and packing dimensions for certain inhomogeneous Moran set constructions. These constructions are beyond the classical theory of iterated function systems, as different nonlinear contraction transformations are applied at each step. Moreover, we also allow the contractions to be weakly conformal and consider situations where the contraction rates have an infimum of zero. In addition, the basic sets of the construction are allowed to have a complicated topology such as having fractal boundaries. Using techniques from thermodynamic formalism we calculate the fractal dimension of the limit set of the construction. As a main application we consider dimension results for stochastic inhomogeneous Moran set constructions, where chaotic dynamical systems are used to control the contraction factors at each step of the construction.

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arXiv2005

Livsic Regularity For Markov Systems

We prove measurable Livsic theorems for dynamical systems modelled by Markov Towers. Our regularity results apply to solutions of cohomological equations posed on Henon-like mappings and a wide variety of nonuniformly hyperbolic systems. We consider both Holder cocycles and cocycles with singularities of prescribed order.

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arXiv2003

A new proof of the Stable Manifold Theorem for hyperbolic fixed points on surfaces

We introduce a new technique for proving the classical Stable Manifold theorem for hyperbolic fixed points. This method is much more geometrical than the standard approaches which rely on abstract fixed point theorems. It is based on the convergence of a canonical sequence of ``finite time local stable manifolds'' which are related to the dynamics of a finite number of iterations.

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