arXiv subjects

Nicolai Haydn

Publications and source records attributed to Nicolai Haydn.

18 recordsLinked to original sources

arXiv2024

Compound Poisson distributions for random dynamical systems using probabilistic approximations

We obtain quenched hitting distributions to be compound Poissonian for a certain class of random dynamical systems. The theory is general and designed to accommodate non-uniformly expanding behavior and targets that do not overlap much with the region where uniformity breaks. Based on annealed and quenched polynomial decay of correlations, our quenched result adopts annealed Kac-type time-normalization and finds limits to be noise-independent. The technique involves a probabilistic blockapproximation where the quenched hit-counting function up to annealed Kac-normalized time is split into equally sized blocks which are mimicked by an independency of random variables distributed just like each of them. The theory is made operational due to a result that allows certain hitting quantities to be recovered from return quantities. Our application is to a class of random piecewise expanding one-dimensional systems, casting new light on the well-known deterministic dichotomy between periodic and aperiodic points, their usual extremal index formula EI=1-1/JT^p(x_0) , and recovering the PolyaAeppli case for general Bernoulli-driven systems, but distinct behavior otherwise. Future and on-going investigations aim to produce and accommodate examples of bonafide nonuniformly expanding random systems and targets approaching their neutral points.

math.DS↗

arXiv2021

Number of visits in arbitrary sets for $ϕ$-mixing dynamics

It is well-known that, for sufficiently mixing dynamical systems, the number of visits to balls and cylinders of vanishing measure is approximately Poisson compound distributed in the Kac scaling. Here we extend this kind of results when the target set is an arbitrary set with vanishing measure in the case of $ϕ$-mixing systems. The error of approximation in total variation is derived using Stein-Chen method. An important part of the paper is dedicated to examples to illustrate the assumptions, as well as applications to temporal synchronisation of $g$-measures

math.DS↗

arXiv2021

Erdos-Renyi laws for exponentially and polynomially mixing dynamical systems

Erdos-Renyi limit laws give the length scale of a time-window over which time-averages in Birkhoff sums have a non-trivial almost-sure limit. We establish Erdos-Renyi type limit laws for Holder observables on dynamical systems modeled by Young Towers with exponential and polynomial tails. This extends earlier results on Erdos-Renyi limit laws to a broad class of dynamical systems with some degree of hyperbolicity.

math.DS↗

arXiv2020

Escape rate and conditional escape rate from a probabilistic point of view

We prove that for a sequence of nested sets $\{U_n\}$ with $Λ= \cap_n U_n$ a measure zero set, the localized escape rate converges to the extremal index of $Λ$, provided that the dynamical system is $ϕ$-mixing at polynomial speed. We also establish the general equivalence between the local escape rate for entry times and the local escape rate for returns.

math.DS↗

arXiv2020

Extreme Value Theory with Spectral Techniques: application to a simple attractor

We give a brief account of application of extreme value theory in dynamical systems by using perturbation techniques associated to the transfer operator. We will apply it to the baker's map and we will get a precise formula for the extremal index. We will also show that the statistics of the number of visits in small sets is compound Poisson distributed.

math.DS↗

arXiv2019

Exponential Law for Random Maps on Compact Manifolds

We consider random dynamical systems on manifolds modeled by a skew product which have certain geometric properties and whose measures satisfy quenched decay of correlations at a sufficient rate. We prove that the limiting distribution for the hitting and return times to geometric balls are both exponential for almost every realisation. We then apply this result to random $C^2$ maps of the interval and random parabolic maps on the unit interval.

math.DS↗

arXiv2019

Local Escape Rates for $ϕ$-mixing Dynamical Systems

We show that dynamical systems with $ϕ$-mixing measures have local escape rates which are exponential with rate $1$ at non-periodic points and equal to the extremal index at periodic points. We apply this result to equilibrium states on subshifts of finite type, expanding interval maps, Gibbs states on conformal repellers and more generally to Young towers and by extension to all systems that can be modeled by a Young tower.

math.DS↗

arXiv2016

Return times at periodic points in random dynamics

We prove a quenched limiting law for random measures on subshifts at periodic points. We consider a family of measures $\{μ_ω\}_{ω\inΩ}$, where the `driving space' $Ω$ is equipped with a probability measure which is invariant under a transformation $θ$. We assume that the fibred measures $μ_ω$ satisfy a generalised invariance property and are $ψ$-mixing. We then show that for almost every $ω$ the return times to cylinders $A_n$ at periodic points are in the limit compound Poisson distributed for a parameter $\vartheta$ which is given by the escape rate at the periodic point.

math.DS↗

arXiv2016

Poisson Law for returns of Maps on Compact Manifolds

We consider invariant measures of maps on manifolds whose correlations decay at a sufficient rate and which satisfy a geometric contraction property. We then prove the that the limiting distribution of returns to geometric balls is Poissonian. This does not assume an tower construction. The decay of correlations is used to show that the independence generated results in the Poisson distribution for returns that are sufficiently separated. A geometric contraction property is then used to show that short return times have a vanishing contribution to the return times distribution. We then also show that the set of very short returns which are of a small linear order of the logarithm of the radius of the balls has a vanishing measure. We obtain error terms which decay polynomially in the logarithm of the radius. We also obtain a extreme value law for such systems.

math.DS↗

arXiv2015

Entry times distribution for mixing systems

We consider the return times dynamics to Bowen balls for continuous maps on metric spaces which have invariant probability measures with certain mixing properties. These mixing properties are satisfied for instance by systems that allow Young tower constructions. We show that the higher order return times to Bowen balls are in the limit Poisson distributed. We also provide a general result for the asymptotic behavior of the recurrence time for Bowen balls for ergodic systems and those with specification.

math.DS↗

arXiv2014

Entry times distribution for dynamical balls on metric spaces

We show that the entry and return times for dynamic balls (Bowen balls) is exponential for systems that have an $α$-mixing invariant measure with certain regularities. We also show that systems modeled by Young's tower has exponential hitting time distribution for dynamical balls

math.DS↗

arXiv2014

Example of a Non-standard Extreme Value Law

It has been shown that sufficiently well mixing dynamical systems with positive entropy have extreme value laws which in the limit converge to one of the three standard distributions known for i.i.d. processes, namely Gumbel, Fréchet and Weibull distributions. In this short note we give an example which has a non-standard limiting distribution for its extreme values. Rotations of the circle by irrational numbers are used and it will be shown that the limiting distribution is a step function where the limit has to be taken along a suitable sequence given by the convergents.

math.PR↗

arXiv2013

Convergence of Rare Events Point Processes to the Poisson for billiards

We show that for planar dispersing billiards the return times distribution is, in the limit, Poisson for metric balls almost everywhere w.r.t. the SRB measure. Since the Poincaré return map is piecewise smooth but becomes singular at the boundaries of the partition elements, recent results on the limiting distribution of return times cannot be applied as they require the maps to have bounded second derivatives everywhere. We first prove the Poisson limiting distribution assuming exponentially decaying correlations. For the case when the correlations decay polynomially, we induce on a subset on which the induced map has exponentially decaying correlations. We then prove a general theorem according to which the limiting return times statistics of the original map and the induced map are the same.

math.DS↗

arXiv2013

Entrance Time and Rényi Entropy

For ergodic systems with generating partitions, the well known result of Ornstein and Weiss shows that the exponential growth rate of the recurrence time is almost surely equal to the metric entropy. Here we look at the exponential growth rate of entrance times, and show that it equals the entropy, where the convergence is in probability in the product measure. This is however under the assumptions that the limiting entrance times distribution exists almost surely. This condition looks natural in the light of an example by Shields in which the limsup in the exponential growth rate is infinite almost everywhere but where the limiting entrance times do not exist. We then also consider $ϕ$-mixing systems and prove a result connecting the Rényi entropy to sums over the entrance times orbit segments.

math.DS↗

arXiv2013

Central limit theorems for the shrinking target problem

Suppose $B_i:= B(p,r_i)$ are nested balls of radius $r_i$ about a point $p$ in a dynamical system $(T,X,μ)$. The question of whether $T^i x\in B_i$ infinitely often (i. o.) for $μ$ a.e.\ $x$ is often called the shrinking target problem. In many dynamical settings it has been shown that if $E_n:=\sum_{i=1}^n μ(B_i)$ diverges then there is a quantitative rate of entry and $\lim_{n\to \infty} \frac{1}{E_n} \sum_{j=1}^{n} 1_{B_i} (T^i x) \to 1$ for $μ$ a.e. $x\in X$. This is a self-norming type of strong law of large numbers. We establish self-norming central limit theorems (CLT) of the form $\lim_{n\to \infty} \frac{1}{a_n} \sum_{i=1}^{n} [1_{B_i} (T^i x)-μ(B_i)] \to N(0,1)$ (in distribution) for a variety of hyperbolic and non-uniformly hyperbolic dynamical systems, the normalization constants are $a^2_n \sim E [\sum_{i=1}^n 1_{B_i} (T^i x)-μ(B_i)]^2$. Dynamical systems to which our results apply include smooth expanding maps of the interval, Rychlik type maps, Gibbs-Markov maps, rational maps and, in higher dimensions, piecewise expanding maps. For such central limit theorems the main difficulty is to prove that the non-stationary variance has a limit in probability.

math.DS↗

arXiv2012

Statistical properties of coupled expanding maps on a lattice with general infinite range couplings and Hölder densities

We continue the development of transfer operator techniques for expanding maps on a lattice coupled by general interaction functions. We obtain a spectral gap for an appropriately defined transfer operator, and, as corollaries, the existence of an invariant conformal probability measure for the system, exponential decay of correlations, the central limit theorem and the almost sure invariance principle.

math.DS↗

arXiv2008

Statistical properties of intermittent maps with unbounded derivative

We study the ergodic and statistical properties of a class of maps of the circle and of the interval of Lorenz type which present indifferent fixed points and points with unbounded derivative. These maps have been previously investigated in the physics literature. We prove in particular that correlations decay polynomially, and that suitable Limit Theorems (convergence to Stable Laws or Central Limit Theorem) hold for Hölder continuous observables. We moreover show that the return and hitting times are in the limit exponentially distributed.

math.DS↗

arXiv1995

Topological entropy of generalized polygon exchanges

We obtain geometric upper bounds on the topological entropy of generalized polygon exchange transformations. As an application of our results, we show that billiards in polygons and rational polytopes have zero topological entropy.

math.DS↗