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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.14297">arXiv:2411.14297</a> <span> [<a href="https://arxiv.org/pdf/2411.14297">pdf</a>, <a href="https://arxiv.org/format/2411.14297">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Chaotic Dynamics">nlin.CD</span> </div> </div> <p class="title is-5 mathjax"> Limitations of the Generalized Pareto Distribution-based estimators for the local dimension </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=del+Amo%2C+I">Ignacio del Amo</a>, <a href="/search/math?searchtype=author&query=Datseris%2C+G">George Datseris</a>, <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.14297v2-abstract-short" style="display: inline;"> 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… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14297v2-abstract-full').style.display = 'inline'; document.getElementById('2411.14297v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.14297v2-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14297v2-abstract-full').style.display = 'none'; document.getElementById('2411.14297v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.13300">arXiv:2401.13300</a> <span> [<a href="https://arxiv.org/pdf/2401.13300">pdf</a>, <a href="https://arxiv.org/ps/2401.13300">ps</a>, <a href="https://arxiv.org/format/2401.13300">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> On distributional limit laws for recurrence </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Todd%2C+M">Mike Todd</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2401.13300v1-abstract-short" style="display: inline;"> For a probability measure preserving dynamical system $(\mathcal{X},f,渭)$, the Poincar茅 Recurrence Theorem asserts that $渭$-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 tim… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.13300v1-abstract-full').style.display = 'inline'; document.getElementById('2401.13300v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.13300v1-abstract-full" style="display: none;"> For a probability measure preserving dynamical system $(\mathcal{X},f,渭)$, the Poincar茅 Recurrence Theorem asserts that $渭$-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 $r_n(蟿)$ follows an \emph{averaged} Poisson distribution $G(蟿)$. Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process $X_n$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.13300v1-abstract-full').style.display = 'none'; document.getElementById('2401.13300v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37A50; 37B20; 60G55; 37E05; 60G70 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.02864">arXiv:2311.02864</a> <span> [<a href="https://arxiv.org/pdf/2311.02864">pdf</a>, <a href="https://arxiv.org/format/2311.02864">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Runs of Extremes of Observables on Dynamical Systems and Applications </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Carney%2C+M">Meagan Carney</a>, <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Nicol%2C+M">Matthew Nicol</a>, <a href="/search/math?searchtype=author&query=Tran%2C+P">Phuong Tran</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.02864v1-abstract-short" style="display: inline;"> 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 Fr茅chet (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-w… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.02864v1-abstract-full').style.display = 'inline'; document.getElementById('2311.02864v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.02864v1-abstract-full" style="display: none;"> 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 Fr茅chet (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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.02864v1-abstract-full').style.display = 'none'; document.getElementById('2311.02864v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2109.06314">arXiv:2109.06314</a> <span> [<a href="https://arxiv.org/pdf/2109.06314">pdf</a>, <a href="https://arxiv.org/ps/2109.06314">ps</a>, <a href="https://arxiv.org/format/2109.06314">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Dichotomy results for eventually always hitting time statistics and almost sure growth of extremes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Kirsebom%2C+M">Maxim Kirsebom</a>, <a href="/search/math?searchtype=author&query=Kunde%2C+P">Philipp Kunde</a>, <a href="/search/math?searchtype=author&query=Persson%2C+T">Tomas Persson</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2109.06314v1-abstract-short" style="display: inline;"> Suppose $(f,\mathcal{X},渭)$ is a measure preserving dynamical system and $蠁\colon \mathcal{X} \to \mathbb{R}$ a measurable function. Consider the maximum process $M_n:=\max\{X_1 \ldots,X_n\}$, where $X_i=蠁\circ f^{i-1}$ is a time series of observations on the system. Suppose that $(u_n)$ is a non-decreasing sequence of real numbers, such that $渭(X_1>u_n)\to 0$. For certain dynamical systems, we ob… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.06314v1-abstract-full').style.display = 'inline'; document.getElementById('2109.06314v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2109.06314v1-abstract-full" style="display: none;"> Suppose $(f,\mathcal{X},渭)$ is a measure preserving dynamical system and $蠁\colon \mathcal{X} \to \mathbb{R}$ a measurable function. Consider the maximum process $M_n:=\max\{X_1 \ldots,X_n\}$, where $X_i=蠁\circ f^{i-1}$ is a time series of observations on the system. Suppose that $(u_n)$ is a non-decreasing sequence of real numbers, such that $渭(X_1>u_n)\to 0$. For certain dynamical systems, we obtain a zero--one measure dichotomy for $渭(M_n\leq u_n\,\textrm{i.o.})$ 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.06314v1-abstract-full').style.display = 'none'; document.getElementById('2109.06314v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">56 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37E05; 37A50; 37D05; 60G70; 11J70 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2108.09395">arXiv:2108.09395</a> <span> [<a href="https://arxiv.org/pdf/2108.09395">pdf</a>, <a href="https://arxiv.org/format/2108.09395">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> On the trajectory of the nonlinear pendulum: Exact analytic solutions via power series </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Reinberger%2C+W+C">W. Cade Reinberger</a>, <a href="/search/math?searchtype=author&query=Holland%2C+M+S">Morgan S. Holland</a>, <a href="/search/math?searchtype=author&query=Barlow%2C+N+S">Nathaniel S. Barlow</a>, <a href="/search/math?searchtype=author&query=Weinstein%2C+S+J">Steven J. Weinstein</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2108.09395v2-abstract-short" style="display: inline;"> We provide an exact infinite power series solution that describes the trajectory of a nonlinear simple pendulum undergoing librating and rotating motion for all time. Although the series coefficients were previously given in [V. Fair茅n, V. L贸pez, and L. Conde, Am. J. Phys 56 (1), (1988), pp. 57-61], the series itself -- as well as the optimal location about which an expansion should be chosen to a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.09395v2-abstract-full').style.display = 'inline'; document.getElementById('2108.09395v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2108.09395v2-abstract-full" style="display: none;"> We provide an exact infinite power series solution that describes the trajectory of a nonlinear simple pendulum undergoing librating and rotating motion for all time. Although the series coefficients were previously given in [V. Fair茅n, V. L贸pez, and L. Conde, Am. J. Phys 56 (1), (1988), pp. 57-61], the series itself -- as well as the optimal location about which an expansion should be chosen to assure series convergence and maximize the domain of convergence -- was not examined, and is provided here. By virtue of its representation as an elliptic function, the pendulum function has singularities that lie off of the real axis in the complex time plane. This, in turn, imposes a radius of convergence on the physical problem in real time. By choosing the expansion point at the top of the trajectory, the power series converges all the way to the bottom of the trajectory without being affected by these singularities. In constructing the series solution, we re-derive the coefficients using an alternative approach that generalizes to other nonlinear problems of mathematical physics. Additionally, we provide an exact resummation of the pendulum series -- Motivated by the asymptotic approximant method given in [Barlow et al., Q. J. Mech. Appl. Math., 70 (1) (2017), pp. 21-48] -- that accelerates the series' convergence uniformly from the top to the bottom of the trajectory. We also provide an accelerated exact resummation of the infinite series representation for the elliptic integral used in calculating the period of a pendulum's trajectory. This allows one to preserve analyticity in the use of the period to extend the pendulum series for all time via symmetry. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.09395v2-abstract-full').style.display = 'none'; document.getElementById('2108.09395v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">The only change in this version is the removal of LaTeX characters in the abstract</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1910.03464">arXiv:1910.03464</a> <span> [<a href="https://arxiv.org/pdf/1910.03464">pdf</a>, <a href="https://arxiv.org/ps/1910.03464">ps</a>, <a href="https://arxiv.org/format/1910.03464">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Limit theorems for wobbly interval intermittent maps </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Coates%2C+D">Douglas Coates</a>, <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Terhesiu%2C+D">Dalia Terhesiu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1910.03464v3-abstract-short" style="display: inline;"> 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 H枚lder observables fail. We obtain limit laws for such maps and H枚lder observables. These limit laws are similar to the classical semistable laws previously established for random processes, but certain limitations imposed by the curren… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.03464v3-abstract-full').style.display = 'inline'; document.getElementById('1910.03464v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1910.03464v3-abstract-full" style="display: none;"> 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 H枚lder observables fail. We obtain limit laws for such maps and H枚lder 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.03464v3-abstract-full').style.display = 'none'; document.getElementById('1910.03464v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 October, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Minor modifications and several typos corrected</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1909.04748">arXiv:1909.04748</a> <span> [<a href="https://arxiv.org/pdf/1909.04748">pdf</a>, <a href="https://arxiv.org/format/1909.04748">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Extremes and extremal indices for level set observables on hyperbolic systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Carney%2C+M">Meagan Carney</a>, <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Nicol%2C+M">Matthew Nicol</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1909.04748v2-abstract-short" style="display: inline;"> Consider an ergodic measure preserving dynamical system $(T,X,渭)$, and an observable $蠁:X\to\mathbb{R}$. For the time series $X_n(x)=蠁(T^{n}(x))$, we establish limit laws for the maximum process $M_n=\max_{k\leq n}X_k$ in the case where $蠁$ is an observable maximized on a curve or submanifold, and $(T,X,渭)$ is a hyperbolic dynamical system. Such observables arise naturally in weather and climate a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.04748v2-abstract-full').style.display = 'inline'; document.getElementById('1909.04748v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1909.04748v2-abstract-full" style="display: none;"> Consider an ergodic measure preserving dynamical system $(T,X,渭)$, and an observable $蠁:X\to\mathbb{R}$. For the time series $X_n(x)=蠁(T^{n}(x))$, we establish limit laws for the maximum process $M_n=\max_{k\leq n}X_k$ in the case where $蠁$ is an observable maximized on a curve or submanifold, and $(T,X,渭)$ 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.04748v2-abstract-full').style.display = 'none'; document.getElementById('1909.04748v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 May, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37A50; 60G70 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1906.10300">arXiv:1906.10300</a> <span> [<a href="https://arxiv.org/pdf/1906.10300">pdf</a>, <a href="https://arxiv.org/format/1906.10300">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Distribution-robust mean estimation via smoothed random perturbations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Holland%2C+M+J">Matthew J. Holland</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1906.10300v1-abstract-short" style="display: inline;"> We consider the problem of mean estimation assuming only finite variance. We study a new class of mean estimators constructed by integrating over random noise applied to a soft-truncated empirical mean estimator. For appropriate choices of noise, we show that this can be computed in closed form, and utilizing relative entropy inequalities, these estimators enjoy deviations with exponential tails c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.10300v1-abstract-full').style.display = 'inline'; document.getElementById('1906.10300v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1906.10300v1-abstract-full" style="display: none;"> We consider the problem of mean estimation assuming only finite variance. We study a new class of mean estimators constructed by integrating over random noise applied to a soft-truncated empirical mean estimator. For appropriate choices of noise, we show that this can be computed in closed form, and utilizing relative entropy inequalities, these estimators enjoy deviations with exponential tails controlled by the second moment of the underlying distribution. We consider both additive and multiplicative noise, and several noise distribution families in our analysis. Furthermore, we empirically investigate the sensitivity to the mean-standard deviation ratio for numerous concrete manifestations of the estimator class of interest. Our main take-away is that an inexpensive new estimator can achieve nearly sub-Gaussian performance for a wide variety of data distributions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.10300v1-abstract-full').style.display = 'none'; document.getElementById('1906.10300v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.10742">arXiv:1810.10742</a> <span> [<a href="https://arxiv.org/pdf/1810.10742">pdf</a>, <a href="https://arxiv.org/format/1810.10742">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Galatolo%2C+S">Stefano Galatolo</a>, <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Persson%2C+T">Tomas Persson</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yiwei Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1810.10742v3-abstract-short" style="display: inline;"> 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 $(f,X,渭)$ preserving a $蟽$-finite measure $渭$ such that $渭(X)=\infty$; ii) the case where $渭$ is a probability measure but we consider the statistical behaviour of an observable $蠁\colon X\to[0,\infty)$ which is non-integrabl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.10742v3-abstract-full').style.display = 'inline'; document.getElementById('1810.10742v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.10742v3-abstract-full" style="display: none;"> 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 $(f,X,渭)$ preserving a $蟽$-finite measure $渭$ such that $渭(X)=\infty$; ii) the case where $渭$ is a probability measure but we consider the statistical behaviour of an observable $蠁\colon X\to[0,\infty)$ which is non-integrable: $\int 蠁\, d渭=\infty$. 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 $蠁$, the local dimension of $渭$, 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.10742v3-abstract-full').style.display = 'none'; document.getElementById('1810.10742v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 April, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">1 figure, 46 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37A40; 37A50; 37A25; 60G75; 37D25 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1608.05620">arXiv:1608.05620</a> <span> [<a href="https://arxiv.org/pdf/1608.05620">pdf</a>, <a href="https://arxiv.org/ps/1608.05620">ps</a>, <a href="https://arxiv.org/format/1608.05620">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Weak convergence to extremal processes and record events for non-uniformly hyperbolic dynamical systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Todd%2C+M">Mike Todd</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1608.05620v2-abstract-short" style="display: inline;"> For a measure preserving dynamical system $(\mathcal{X},f, 渭)$, we consider the time series of maxima $M_n=\max\{X_1,\ldots,X_n\}$ associated to the process $X_n=蠁(f^{n-1}(x))$ generated by the dynamical system for some observable $蠁:\mathcal{X}\to\mathbb{R}$. 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 su… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1608.05620v2-abstract-full').style.display = 'inline'; document.getElementById('1608.05620v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1608.05620v2-abstract-full" style="display: none;"> For a measure preserving dynamical system $(\mathcal{X},f, 渭)$, we consider the time series of maxima $M_n=\max\{X_1,\ldots,X_n\}$ associated to the process $X_n=蠁(f^{n-1}(x))$ generated by the dynamical system for some observable $蠁:\mathcal{X}\to\mathbb{R}$. 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1608.05620v2-abstract-full').style.display = 'none'; document.getElementById('1608.05620v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 May, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 August, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">To appear in Ergodic Theory Dynam. Systems</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1605.07006">arXiv:1605.07006</a> <span> [<a href="https://arxiv.org/pdf/1605.07006">pdf</a>, <a href="https://arxiv.org/format/1605.07006">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistical Mechanics">cond-mat.stat-mech</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Chaotic Dynamics">nlin.CD</span> </div> </div> <p class="title is-5 mathjax"> Extremes and Recurrence in Dynamical Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lucarini%2C+V">Valerio Lucarini</a>, <a href="/search/math?searchtype=author&query=Faranda%2C+D">Davide Faranda</a>, <a href="/search/math?searchtype=author&query=Freitas%2C+A+C+M">Ana Cristina Moreira Freitas</a>, <a href="/search/math?searchtype=author&query=Freitas%2C+J+M">Jorge Milhazes Freitas</a>, <a href="/search/math?searchtype=author&query=Kuna%2C+T">Tobias Kuna</a>, <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Nicol%2C+M">Matthew Nicol</a>, <a href="/search/math?searchtype=author&query=Todd%2C+M">Mike Todd</a>, <a href="/search/math?searchtype=author&query=Vaienti%2C+S">Sandro Vaienti</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1605.07006v1-abstract-short" style="display: inline;"> 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 class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.07006v1-abstract-full').style.display = 'inline'; document.getElementById('1605.07006v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1605.07006v1-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.07006v1-abstract-full').style.display = 'none'; document.getElementById('1605.07006v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 May, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">305 pages book, V. Lucarini, D. Faranda, A. C. M. Freitas, J. M. Freitas, T. Kuna, M. Holland, M. Nicol, M. Todd, S. Vaienti, Extremes and Recurrence in Dynamical Systems, Wiley, New York, 2016, ISBN: 978-1-118-63219-2</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60G70; 37A60; 37A25; 62M10; 82C05; 86A04 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1510.04681">arXiv:1510.04681</a> <span> [<a href="https://arxiv.org/pdf/1510.04681">pdf</a>, <a href="https://arxiv.org/ps/1510.04681">ps</a>, <a href="https://arxiv.org/format/1510.04681">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Almost sure convergence of maxima for chaotic dynamical systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Holland%2C+M+P">M. P. Holland</a>, <a href="/search/math?searchtype=author&query=Nicol%2C+M">M. Nicol</a>, <a href="/search/math?searchtype=author&query=T%C3%B6r%C3%B6k%2C+A">A. T枚r枚k</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1510.04681v1-abstract-short" style="display: inline;"> Suppose $(f,\mathcal{X},谓)$ is a measure preserving dynamical system and $蠁:\mathcal{X}\to\mathbb{R}$ is an observable with some degree of regularity. We investigate the maximum process $M_n:=\max\{X_1,\ldots,X_n\}$, where $X_i=蠁\circ f^i$ is a time series of observations on the system. When $M_n\to\infty$ almost surely, we establish results on the almost sure growth rate, namely the existence (or… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1510.04681v1-abstract-full').style.display = 'inline'; document.getElementById('1510.04681v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1510.04681v1-abstract-full" style="display: none;"> Suppose $(f,\mathcal{X},谓)$ is a measure preserving dynamical system and $蠁:\mathcal{X}\to\mathbb{R}$ is an observable with some degree of regularity. We investigate the maximum process $M_n:=\max\{X_1,\ldots,X_n\}$, where $X_i=蠁\circ f^i$ is a time series of observations on the system. When $M_n\to\infty$ almost surely, we establish results on the almost sure growth rate, namely the existence (or otherwise) of a sequence $u_n\to\infty$ such that $M_n/u_n\to 1$ almost surely. The observables we consider will be functions of the distance to a distinguished point $\tilde{x}\in \mathcal{X}$. Our results are based on the interplay between shrinking target problem estimates at $\tilde{x}$ and the form of the observable (in particular polynomial or logarithmic) near $\tilde{x}$. We determine where such an almost sure limit exists and give examples where it does not. Our results apply to a wide class of non-uniformly hyperbolic dynamical systems, under mild assumptions on the rate of mixing, and on regularity of the invariant measure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1510.04681v1-abstract-full').style.display = 'none'; document.getElementById('1510.04681v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 October, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1501.05023">arXiv:1501.05023</a> <span> [<a href="https://arxiv.org/pdf/1501.05023">pdf</a>, <a href="https://arxiv.org/format/1501.05023">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1080/14689367.2015.1056722">10.1080/14689367.2015.1056722 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Extremal dichotomy for uniformly hyperbolic systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Carvalho%2C+M">Maria Carvalho</a>, <a href="/search/math?searchtype=author&query=Freitas%2C+A+C+M">Ana Cristina Moreira Freitas</a>, <a href="/search/math?searchtype=author&query=Freitas%2C+J+M">Jorge Milhazes Freitas</a>, <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Nicol%2C+M">Matthew Nicol</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1501.05023v2-abstract-short" style="display: inline;"> We consider the extreme value theory of a hyperbolic toral automorphism $T: \mathbb{T}^2 \to \mathbb{T}^2$ showing that if a H枚lder observation $蠁$ which is a function of a Euclidean-type distance to a non-periodic point $味$ is strictly maximized at $味$ then the corresponding time series $\{蠁\circ T^i\}$ exhibits extreme value statistics corresponding to an iid sequence of random variables with th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1501.05023v2-abstract-full').style.display = 'inline'; document.getElementById('1501.05023v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1501.05023v2-abstract-full" style="display: none;"> We consider the extreme value theory of a hyperbolic toral automorphism $T: \mathbb{T}^2 \to \mathbb{T}^2$ showing that if a H枚lder observation $蠁$ which is a function of a Euclidean-type distance to a non-periodic point $味$ is strictly maximized at $味$ then the corresponding time series $\{蠁\circ T^i\}$ exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $蠁$ and with extremal index one. If however $蠁$ 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 $蠁$ 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1501.05023v2-abstract-full').style.display = 'none'; document.getElementById('1501.05023v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 January, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 January, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, 4 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37A50; 60G70; 37B20; 60G10; 37C25 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Dyn. Syst., 30(4):38--403, 2015 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1404.3941">arXiv:1404.3941</a> <span> [<a href="https://arxiv.org/pdf/1404.3941">pdf</a>, <a href="https://arxiv.org/ps/1404.3941">ps</a>, <a href="https://arxiv.org/format/1404.3941">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Quantitative recurrence statistics and convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Rabassa%2C+P">Pau Rabassa</a>, <a href="/search/math?searchtype=author&query=Sterk%2C+A">Alef Sterk</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1404.3941v2-abstract-short" style="display: inline;"> 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 r… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1404.3941v2-abstract-full').style.display = 'inline'; document.getElementById('1404.3941v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1404.3941v2-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1404.3941v2-abstract-full').style.display = 'none'; document.getElementById('1404.3941v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 September, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 April, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This article is a revision of the previous article titled: "On the convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems." Relative to this older version, the revised article includes new and up to date results and developments (based upon recent advances in the field)</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1211.0927">arXiv:1211.0927</a> <span> [<a href="https://arxiv.org/pdf/1211.0927">pdf</a>, <a href="https://arxiv.org/ps/1211.0927">ps</a>, <a href="https://arxiv.org/format/1211.0927">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Dimension results for inhomogeneous Moran set constructions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yiwei Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1211.0927v2-abstract-short" style="display: inline;"> 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… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1211.0927v2-abstract-full').style.display = 'inline'; document.getElementById('1211.0927v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1211.0927v2-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1211.0927v2-abstract-full').style.display = 'none'; document.getElementById('1211.0927v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 November, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 November, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30pages and 1 figure</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1203.5145">arXiv:1203.5145</a> <span> [<a href="https://arxiv.org/pdf/1203.5145">pdf</a>, <a href="https://arxiv.org/ps/1203.5145">ps</a>, <a href="https://arxiv.org/format/1203.5145">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> On the mixing properties of piecewise expanding maps under composition with permutations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Byott%2C+N+P">Nigel P. Byott</a>, <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yiwei Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1203.5145v3-abstract-short" style="display: inline;"> 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 $蟽$ for which $蟽\circ f$ is still (topologically)… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1203.5145v3-abstract-full').style.display = 'inline'; document.getElementById('1203.5145v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1203.5145v3-abstract-full" style="display: none;"> 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 $蟽$ for which $蟽\circ f$ 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 $蟽\circ f$ (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 $蟽$ 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1203.5145v3-abstract-full').style.display = 'none'; document.getElementById('1203.5145v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 December, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 March, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2012. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1107.5673">arXiv:1107.5673</a> <span> [<a href="https://arxiv.org/pdf/1107.5673">pdf</a>, <a href="https://arxiv.org/ps/1107.5673">ps</a>, <a href="https://arxiv.org/format/1107.5673">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Chaotic Dynamics">nlin.CD</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Data Analysis, Statistics and Probability">physics.data-an</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physd.2011.11.005">10.1016/j.physd.2011.11.005 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Extreme value laws in dynamical systems under physical observables </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Holland%2C+M+P">Mark P. Holland</a>, <a href="/search/math?searchtype=author&query=Vitolo%2C+R">Renato Vitolo</a>, <a href="/search/math?searchtype=author&query=Rabassa%2C+P">Pau Rabassa</a>, <a href="/search/math?searchtype=author&query=Sterk%2C+A+E">Alef E. Sterk</a>, <a href="/search/math?searchtype=author&query=Broer%2C+H+W">Henk W. Broer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1107.5673v1-abstract-short" style="display: inline;"> Extreme value theory for chaotic dynamical systems is a rapidly expanding area of research. Given a system and a real function (observable) defined on its phase space, extreme value theory studies the limit probabilistic laws obeyed by large values attained by the observable along orbits of the system. Based on this theory, the so-called block maximum method is often used in applications for stati… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1107.5673v1-abstract-full').style.display = 'inline'; document.getElementById('1107.5673v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1107.5673v1-abstract-full" style="display: none;"> Extreme value theory for chaotic dynamical systems is a rapidly expanding area of research. Given a system and a real function (observable) defined on its phase space, extreme value theory studies the limit probabilistic laws obeyed by large values attained by the observable along orbits of the system. Based on this theory, the so-called block maximum method is often used in applications for statistical prediction of large value occurrences. In this method, one performs inference for the parameters of the Generalised Extreme Value (GEV) distribution, using maxima over blocks of regularly sampled observations along an orbit of the system. The observables studied so far in the theory are expressed as functions of the distance with respect to a point, which is assumed to be a density point of the system's invariant measure. However, this is not the structure of the observables typically encountered in physical applications, such as windspeed or vorticity in atmospheric models. In this paper we consider extreme value limit laws for observables which are not functions of the distance from a density point of the dynamical system. In such cases, the limit laws are no longer determined by the functional form of the observable and the dimension of the invariant measure: they also depend on the specific geometry of the underlying attractor and of the observable's level sets. We present a collection of analytical and numerical results, starting with a toral hyperbolic automorphism as a simple template to illustrate the main ideas. We then formulate our main results for a uniformly hyperbolic system, the solenoid map. We also discuss non-uniformly hyperbolic examples of maps (H茅non and Lozi maps) and of flows (the Lorenz63 and Lorenz84 models). Our purpose is to outline the main ideas and to highlight several serious problems found in the numerical estimation of the limit laws. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1107.5673v1-abstract-full').style.display = 'none'; document.getElementById('1107.5673v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 July, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2011. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0709.1723">arXiv:0709.1723</a> <span> [<a href="https://arxiv.org/pdf/0709.1723">pdf</a>, <a href="https://arxiv.org/format/0709.1723">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Statistical properties of one-dimensional maps with critical points and singularities </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=D%C3%ADaz-Ordaz%2C+K">K. D铆az-Ordaz</a>, <a href="/search/math?searchtype=author&query=Holland%2C+M+P">M. P. Holland</a>, <a href="/search/math?searchtype=author&query=Luzzatto%2C+S">S. Luzzatto</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0709.1723v1-abstract-short" style="display: inline;"> We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for H枚lder observations. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0709.1723v1-abstract-full" style="display: none;"> We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for H枚lder observations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0709.1723v1-abstract-full').style.display = 'none'; document.getElementById('0709.1723v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 September, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2007. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D50; 37A25 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Stochastics and Dynamics, vol 6, issue 4 (2006), pp. 423-458 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0503690">arXiv:math/0503690</a> <span> [<a href="https://arxiv.org/pdf/math/0503690">pdf</a>, <a href="https://arxiv.org/ps/math/0503690">ps</a>, <a href="https://arxiv.org/format/math/0503690">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Livsic Regularity For Markov Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bruin%2C+H">Henk Bruin</a>, <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Nicol%2C+M">Matthew Nicol</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0503690v2-abstract-short" style="display: inline;"> 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. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0503690v2-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0503690v2-abstract-full').style.display = 'none'; document.getElementById('math/0503690v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 August, 2005; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 March, 2005; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2005. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">35 pages, To be published in Ergodic Theory and Dynamical Systems</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37A29 (Primary) 37C15; 37D25 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0502236">arXiv:math/0502236</a> <span> [<a href="https://arxiv.org/pdf/math/0502236">pdf</a>, <a href="https://arxiv.org/ps/math/0502236">ps</a>, <a href="https://arxiv.org/format/math/0502236">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jde.2005.07.013">10.1016/j.jde.2005.07.013 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Stable manifolds under very weak hyperbolicity conditions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Luzzatto%2C+S">Stefano Luzzatto</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0502236v1-abstract-short" style="display: inline;"> We present an argument for proving the existence of local stable and unstable manifolds in a general abstract setting and under very weak hyperbolicity conditions. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0502236v1-abstract-full" style="display: none;"> We present an argument for proving the existence of local stable and unstable manifolds in a general abstract setting and under very weak hyperbolicity conditions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0502236v1-abstract-full').style.display = 'none'; document.getElementById('math/0502236v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 February, 2005; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2005. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D99; 37E30 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Differential Equations 221 (2006) no.2, 444-469 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0301235">arXiv:math/0301235</a> <span> [<a href="https://arxiv.org/pdf/math/0301235">pdf</a>, <a href="https://arxiv.org/ps/math/0301235">ps</a>, <a href="https://arxiv.org/format/math/0301235">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> A new proof of the Stable Manifold Theorem for hyperbolic fixed points on surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Holland%2C+M">Mark Holland</a>, <a href="/search/math?searchtype=author&query=Luzzatto%2C+S">Stefano Luzzatto</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0301235v1-abstract-short" style="display: inline;"> 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. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0301235v1-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0301235v1-abstract-full').style.display = 'none'; document.getElementById('math/0301235v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 January, 2003; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2003. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">1 Figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D10 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: 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