Non-uniformly Hyperbolic Dynamical Systems Research Articles

Local large deviations for periodic infinite horizon Lorentz gases

AbstractWe prove optimal local large deviations for the periodic infinite horizon Lorentz gas viewed as a ℤd-cover (d = 1,2) of a dispersing billiard. In addition to this specific example, we prove a general result for a class of nonuni-formly hyperbolic dynamical systems and observables associated with central limit theorems with nonstandard normalisation.

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

For a measure-preserving dynamical system $({\mathcal{X}},f,\unicode[STIX]{x1D707})$, we consider the time series of maxima $M_{n}=\max \{X_{1},\ldots ,X_{n}\}$ associated to the process $X_{n}=\unicode[STIX]{x1D719}(f^{n-1}(x))$ generated by the dynamical system for some observable $\unicode[STIX]{x1D719}:{\mathcal{X}}\rightarrow \mathbb{R}$. 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 takes 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.

Multifractal analysis of the historic set in non-uniformly hyperbolic systems

We get the exact Hausdorff dimension of the historic set for ratios of the Birkhoff average in a class of one dimensional non-uniformly hyperbolic dynamical systems.

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.

Almost sure convergence of maxima for chaotic dynamical systems

Suppose (f,X,ν) is a measure preserving dynamical system and ϕ:X→R is an observable with some degree of regularity. We investigate the maximum process Mn:=max(X1,…,Xn), where Xi=ϕ∘fi is a time series of observations on the system. When Mn→∞ almost surely, we establish results on the almost sure growth rate, namely the existence (or otherwise) of a sequence un→∞ such that Mn/un→1 almost surely. For a wide class of non-uniformly hyperbolic dynamical systems we determine where such an almost sure limit exists and give examples where it does not.

Speed of convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems

Suppose (f, 𝒳, ν) is a dynamical system and ϕ : 𝒳 → ℝ is an observation with a unique maximum at a (generic) point in 𝒳. We consider the time series of successive maxima Mn(x) := max {ϕ(x),…,ϕ ◦ fn-1(x)}. Recent works have focused on the distributional convergence of such maxima (under suitable normalization) to an extreme value distribution. In this paper, for certain dynamical systems, we establish convergence rates to the limiting distribution. In contrast to the case of i.i.d. random variables, the convergence rates depend on the rate of mixing and the recurrence time statistics. For a range of applications, including uniformly expanding maps, quadratic maps, and intermittent maps, we establish corresponding convergence rates. We also establish convergence rates for certain hyperbolic systems such as Anosov systems, and discuss convergence rates for non-uniformly hyperbolic systems, such as Hénon maps.

Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems

We show that for systems that allow a Young tower construction with polynomially decaying correlations the return times to metric balls are in the limit Poisson distributed. We also provide error terms which are powers of logarithm of the radius. In order to get those uniform rates of convergence the balls centres have to avoid a set whose size is estimated to be of similar order. This result can be applied to non-uniformly hyperbolic maps and to any invariant measure that satisfies a weak regularity condition. In particular it shows that the return times to balls is Poissonian for SRB measures on attractors.

Poisson law for some non-uniformly hyperbolic dynamical systems with polynomial rate of mixing

We consider some non-uniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs–Markov–Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball$B(x,r)$converges to a Poisson distribution as the radius$r\rightarrow 0$and after suitable normalization.

Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems

Nous considérons des principes d’invariance faibles (théorèmes limites fonctionnels) dans le domaine d’une loi stable. Un résultat général est obtenu en relevant de telles lois limites depuis un système dynamique induit vers le système original. Une classe importante d’exemples couverte par notre résultat est donnée par les transformations intermittentes à la Pomeau–Manneville, où la convergence pour le système induit est dans la topologie $\mathcal{J}_{1}$ de Skorohod standard. Pour le système complet, il n’y a pas de convergence dans la topologie $\mathcal{J}_{1}$, mais nous prouvons la convergence dans la topologie $\mathcal{M}_{1}$.

From local to global equilibrium states: Thermodynamic formalism via an inducing scheme

We present a method to construct equilibrium states via inducing. This method can be used for some non-uniformly hyperbolic dynamical systems and for non-Hölder continuous potentials. It allows us to prove the existence of phase transition.

Intrinsic mode functions locate implicit turbulent attractors in time in frontal lobe MEG recordings

In seeking evidence for the presence and characteristic range of coupled time scale(s) of putative implicit turbulent attractors of dorsal frontal lobe magnetic fields, the recorded nonstationary, nonlinear MEG signals were non-orthogonally decomposed using Huang’s Empirical Mode Decomposition, EMD, (Huang and Attoh-Okine, 2005) into 16 Intrinsic Mode Functions, EMD→IMFi, i=1…16. Measures known to be invariant in non-uniformly hyperbolic (turbulent) dynamical systems, topological entropy, hT, metric entropy, hM, non-uniform entropy, hU and power spectral scaling exponent, α, were imposed on each of the IMFi which evidenced most clearly an invariant temporal scale zone of IMFi, i=6…11, for hT, which we have found to be the most robust of invariant measures of MEG’s magnetic field turbulent attractors (Mandell et al., 2011a,b; Mandell, 2013). The ergodic theory of dynamical systems (Walters, 1982; Pollicott and Yuri, 1998) allows the inference that an implicit attractor with consistently hT>0 will also evidence at least one positive Lyapounov exponent indicating the presence of a turbulent attractor with exponential separation of nearby initial conditions, exponential convergence of distant points and disordering, mixing, of orbital sequences. It appears that this approach permits the inference of the presence of chaotic, turbulent attractor and its characteristic time scales without the invocation of arbitrary n-dimensional embedding, phase space reconstructions or (inappropriate) orthogonal decompositions.

The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.

A note on Borel–Cantelli lemmas for non-uniformly hyperbolic dynamical systems

AbstractLet (Bi) be a sequence of measurable sets in a probability space (X,ℬ,μ) such that ∑ ∞n=1μ(Bi)=∞. The classical Borel–Cantelli lemma states that if the sets Bi are independent, then μ({x∈X:x∈Bi infinitely often})=1. Suppose (T,X,μ) is a dynamical system and (Bi) is a sequence of sets in X. We consider whether Tix∈Bi infinitely often for μ almost every x∈X and, if so, is there an asymptotic estimate on the rate of entry? If Tix∈Bi infinitely often for μ almost every x, we call the sequence (Bi) a Borel–Cantelli sequence. If the sets Bi :=B(p,ri) are nested balls of radius ri about a point p, then the question of whether Tix∈Bi infinitely often for μ almost every x is often called the shrinking target problem. We show, under certain assumptions on the measure μ, that for balls Bi if μ(Bi)≥i−γ, 0<γ<1, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observables implies that the sequence is Borel–Cantelli. If μ(Bi)≥C1 /i, then exponential decay of correlations implies that the sequence is Borel–Cantelli. We give conditions in terms of return time statistics which quantify Borel–Cantelli results for sequences of balls such that μ(Bi)≥C/i. Corollaries of our results are that for planar dispersing billiards and Lozi maps, sequences of nested balls B(p,1/i) are Borel–Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems.

Poisson approximation for the number of visits to balls in non-uniformly hyperbolic dynamical systems

AbstractWe study the number of visits to balls Br(x), up to time t/μ(Br(x)), for a class of non-uniformly hyperbolic dynamical systems, where μ is the Sinai–Ruelle–Bowen measure. Outside a set of ‘bad’ centers x, we prove that this number is approximately Poissonnian with a controlled error term. In particular, when r→0, we get convergence to the Poisson law for a set of centers of μ-measure one. Our theorem applies for instance to the Hénon attractor and, more generally, to systems modelled by a Young tower whose return-time function has an exponential tail and with one-dimensional unstable manifolds. Along the way, we prove an abstract Poisson approximation result of independent interest.

Open problems in the theory of non-uniform hyperbolicity

This is a survey-type article whose goal is to review some recent developments in studying the genericity problem for non-uniformly hyperbolic dynamical systems with discrete time on compact smooth manifolds. We discuss both cases of systems which are conservative (preserve the Riemannian volume) and dissipative (possess hyperbolic attractors). We also consider the problem of coexistence of hyperbolic and regular behaviour.

A vector-valued almost sure invariance principle for hyperbolic dynamical systems

We prove an almost sure invariance principle (approximation by d-dimensional Brownian motion) for vector-valued Holder observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom A dieomorphisms and flows as well as systems modelled by Young towers with moderate tail decay rates. In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a 2-dimensional Brownian motion.

Large and moderate deviations for slowly mixing dynamical systems

We obtain results on large deviations for a large class of nonuniformly hyperbolic dynamical systems with polynomial decay of correlations1/nβ1/n^\beta,β>0\beta >0. This includes systems modelled by Young towers with polynomial tails, extending recent work of M. Nicol and the author which assumedβ>1\beta >1. As a byproduct of the proof, we obtain slightly stronger results even whenβ>1\beta >1. The results are sharp in the sense that there exist examples (such as Pomeau-Manneville intermittency maps) for which the obtained rates are best possible. In addition, we obtain results on moderate deviations.

Large deviations in non-uniformly hyperbolic dynamical systems

AbstractWe prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems

In this paper, we apply the Devroye inequality to study various statistical estimators and fluctuations of observables for processes. Most of these observables are suggested by dynamical systems. These applications concern the co-variance function, the integrated periodogram, the correlation dimension, the kernel density estimator, the speed of convergence of the empirical measure, the shadowing property and the almost-sure central limit theorem. We proved (Chazottes et al 2005 Nonlinearity 18 2323–40) that the Devroye inequality holds for a class of non-uniformly hyperbolic dynamical systems introduced in Young (1998 Ann. Math. 147 585–650). In the second appendix we prove that, if the decay of correlations holds with a common rate for all pairs of functions, then it holds uniformly in the function spaces. In the last appendix we prove that, for the subclass of one-dimensional systems studied in Young, the density of the absolutely continuous invariant measure belongs to a Besov space.

Devroye inequality for a class of non-uniformly hyperbolic dynamical systems

In this paper we prove an inequality which we call the ‘Devroye inequality’ for a large class of non-uniformly hyperbolic dynamical systems (M, f). This class, introduced by Young, includes families of piecewise hyperbolic maps (Lozi-like maps), scattering billiards (e.g. planar Lorentz gas), unimodal and Hénon-like maps. The Devroye inequality provides an upper bound for the variance of observables of the form K(x, f(x), …, fn−1(x)), where K is any separately Hölder continuous function of n variables. In particular, we can deal with observables which are not Birkhoff averages. We will show in Chazottes et al (2005 Nonlinearity 18 2341–64) some applications of Devroye inequality to statistical properties of this class of dynamical systems.