For non uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds when the reference point is the initial point of the trajectory. We use the framework for non uniformly hyperbolic dynamical systems developed by L.S.Young.
We consider generalized closest return times of a complex polynomial of degree at least two. Most previous studies on this subject have focused on the properties of polynomials with particular return times, especially the Fibonacci numbers. We study the general form of these closest return times. The main result of this paper is that these closest return times are meta-Fibonacci numbers. This result applies to the return times of a principal nest of a polynomial. Furthermore, we show that an analogous result holds in a tree with dynamics that is associated with a polynomial.
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 measure…
We prove that, in Young towers with sufficiently small tails, the speed in the central limit theorem is O(1/\sqrt{n}), and the local limit theorem holds. This implies the same results for many non uniformly expanding dynamical systems, namely those for which a tower with sufficiently fast returns can be constructed.
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 si…
For non-uniformly expanding maps inducing with a general return time to Gibbs Markov maps, we provide sufficient conditions for obtaining higher order asymptotics for the correlation function in the infinite measure setting. Along the way, we show that these conditions are sufficient to recover previous results on sharp mixing rates in the finite measure setting for non-Markov maps, but for a larger class of observables. The results are illustrated by (finite and infinite measure preserving) non-Markov intervals maps with an indifferent fixed point.
We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure for which the decay of the return time function can be controlled in terms of the time generic points need to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem.
In this note, we obtain verifiable sufficient conditions for the extreme value distribution for a certain class of skew product extensions of non-uniformly hyperbolic base maps. We show that these conditions, formulated in terms of the decay of correlations on the product system and the measure of rapidly returning points on the base, lead to a distribution for the maximum of $Φ(p) = -\log(d(p, p_0))$ that is of the first type. In particular, we establish the Type I distribution for $S^1$ extensions of piecewise $C^2$ uniformly expanding maps of the interval, non-uniformly expanding maps of th…
We study decay of correlations, the asymptotic distribution of hitting times and fluctuations of the return times for a robust class of multidimensional non-uniformly hyperbolic transformations. Oliveira and Viana [15] proved that there is a unique equilibrium state for a large class of non- uniformly expanding transformations and Holder continuous potentials with small variation. For an open class of potentials with small variation, we prove quasi-compactness of the Ruelle-Perron-Frobenius operator in a space $V_θ$ of functions with essential bounded variation that strictly contain Holder con…
Results of Wyner and Ziv and of Ornstein and Weiss show that if one observes the first k outputs of a finite-valued ergodic process, then the waiting time until this block appears again is almost surely asymptotic to $2^{hk}$, where $h$ is the entropy of the process. We examine this phenomenon when the allowed return times are restricted to some subset of times, and generalize the results to processes parameterized by other discrete amenable groups. We also obtain a uniform density version of the waiting time results: For a process on $s$ symbols, within a given realization, the density of the…
We prove a Berry-Esseen theorem, a local central limit theorem and (local) large and (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain random (Young) tower extensions, which is the main focus of this paper. On the random towers we will obtain our results using contraction properties of random complex equivariant cones with respect to the complex Hilbert projective metric.
We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism $f$ of a compact manifold $X$ preserving a hyperbolic ergodic probability measure $μ$. The cocycle $A$ over $f$ is Holder continuous and takes values in $GL(d,R)$ or, more generally, in the group of invertible bounded linear operators on a Banach space. For a $GL(d,R)$-valued cocycle $A$ we prove that the Lyapunov exponents of $A$ with respect to $μ$ can be approximated by the Lyapunov exponents of $A$ with respect to measures on hyperbolic periodic orbits of $f$. In the infinite-di…
We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat singularities or criticalities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/critical set of the base map. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the semiflow, showing that the Lebesgue measure of the set of points whose time averages stay away from t…
We give conditions under which nonuniformly expanding maps exhibit lower bounds of polynomial type for the decay of correlations and for a large class of observables. We show that if the Lasota-Yorke type inequality for the transfer operator of a first return map are satisfied in a Banach space B, and the absolutely continuous invariant measure obtained is weak mixing, in terms of aperiodicity, then under some renewal condition, the maps have polynomial decay of correlations for observables in B. We also provide some general conditions that give aperiodicity for expanding maps in higher dimens…
Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure preserving dynamical system is explicitly controlled. In uniformly hyper-bolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows, and counterexamples illustrating the necessi…
We consider the extreme value theory of a hyperbolic toral automorphism $T: \mathbb{T}^2 \to \mathbb{T}^2$ showing that if a Hölder observation $ϕ$ which is a function of a Euclidean-type distance to a non-periodic point $ζ$ is strictly maximized at $ζ$ then the corresponding time series $\{ϕ\circ T^i\}$ exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $ϕ$ and with extremal index one. If however $ϕ$ is strictly maximized at a periodic point $q$ then the corresponding time-series exhibits extreme value statistics corre…
Let $(B_{i})$ be a sequence of measurable sets in a probability space $(X,\mathcal{B}, μ)$ such that $\sum_{n=1}^{\infty} μ(B_{i}) = \infty$. The classical Borel-Cantelli lemma states that if the sets $B_{i}$ are independent, then $μ(\{x \in X : x \in B_{i} \text{infinitely often (i.o.)}) = 1$. Suppose $(T,X,μ)$ is a dynamical system and $(B_i)$ is a sequence of sets in $X$. We consider whether $T^i x\in B_i$ for $μ$ a.e.\ $x\in X$ and if so, is there an asymptotic estimate on the rate of entry. If $T^i x\in B_i$ infinitely often for $μ$ a.e.\ $x$ we call the sequence $B_i$ a Borel--Cantelli s…
We consider a partially hyperbolic set $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the centre-unstable direction $E^{cu}$ expands non-uniformly on some local unstable disk. We show that under these assumptions $f$ induces a Gibbs-Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the centre-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an al…
We show that a uniformly continuous random perturbation of a transitive map defines an aperiodic Harris chain which also satisfies Doeblin's condition. As a result, we get exponential decay of correlations for suitable random perturbations of such systems. We also prove that, for transitive maps, the limiting distribution for Extreme Value Laws (EVLs) and Hitting/Return Time Statistics (HTS/RTS) is standard exponential. Moreover, we show that the Rare Event Point Process (REPP) converges in distribution to a standard Poisson process.
We study the spectral properties of ergodic Schrödinger operators that are associated to a certain family of non-primitive substitutions on a binary alphabet. The corresponding subshifts provide examples of dynamical systems that go beyond minimality, unique ergodicity and linear complexity. In some parameter region, we are naturally in the setting of an infinite ergodic measure. The almost sure spectrum is singular and contains an interval. Some criteria for the exclusion of eigenvalues are fully characterized, including the existence of strongly palindromic sequences. Many of our structural …
We study the number of visits to balls B_r(x), up to time t/mu(B_r(x)), for a class of non-uniformly hyperbolic dynamical systems, where mu is the SRB 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 mu-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 a exponential tail and with one-dimensional unstable manifolds. Along t…
