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A high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities, called an observation. Following this idea and in the continuity of Boshernitzan's work,…

This paper is a first step in the study of the recurrence behavior in random dynamical systems and randomly perturbed dynamical systems. In particular we define a concept of quenched and annealed return times for systems generated by the…

We consider invertible discrete-time dynamical systems having a hyperbolic product structure in some region of the phase space with infinitely many branches and variable recurrence time. We show that the decay of correlations of the SRB…

We consider low--dimensional dynamical systems with a mixed phase space and discuss the typical appearance of slow, polynomial decay of correlations: in particular we emphasize how this mixing rate is related to large deviations properties.

In this paper we study quantitative recurrence and the shrinking target problem for dynamical systems coming from overlapping iterated function systems. Such iterated function systems have the important property that a point often has…

For a family of random intermittent dynamical systems with a superattracting fixed point we prove that a phase transition occurs between the existence of an absolutely continuous invariant probability measure and infinite measure depending…

We claim that looking at probability distributions of \emph{finite time} largest Lyapunov exponents, and more precisely studying their large deviation properties, yields an extremely powerful technique to get quantitative estimates of…

Let $ ([0,1]^d,T,\mu) $ be a measure-preserving dynamical system so that the correlations decay exponentially for H\"older continuous functions. Suppose that $ \mu $ is absolutely continuous with a density function $ h\in L^q(\mathcal L^d)…

We investigate the statistics of recurrences to finite size intervals for chaotic dynamical systems. We find that the typical distribution presents an exponential decay for almost all recurrence times except for a few short times affected…

Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighborhood of the origin. We address this question in…

For a probability measure preserving dynamical system $(\mathcal{X},f,\mu)$, the Poincar\'e Recurrence Theorem asserts that $\mu$-almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics…

We study Poincar\'e recurrence from a purely geometrical viewpoint. We prove that the metric entropy is given by the exponential growth rate of return times to dynamical balls. This is the geometrical counterpart of Ornstein-Weiss theorem.…

Quantitative recurrence indicators are defined by measuring the first entrance time of the orbit of a point $x$ in a decreasing sequence of neighborhoods of another point $y$. It is proved that these recurrence indicators are a.e. greater…

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…

We consider toral extensions of hyperbolic dynamical systems. We prove that its quantitative recurrence (also with respect to given observables) and hitting time scale behavior depend on the arithmetical properties of the extension. By this…

Recurrence problems are fundamental in dynamics, and for example, sizes of the set of points recurring infinitely often to a target have been studied extensively in many contexts. For example, the problem of finding the dimension for…

We study Poincar\'e recurrence for flows and observations of flows. For Anosov flow, we prove that the recurrence rates are linked to the local dimension of the invariant measure. More generally, we give for the recurrence rates for the…

We investigate the dependence of Poincar\'e recurrence-times statistics on the choice of recurrence-set, by sampling the dynamics of two- and four-dimensional Hamiltonian maps. We derive a method that allows us to visualize the direct…

Using a result of Behrend concerning sets without arithmetic progressions, we construct some examples of dynamical systems with slow time of multiple recurrence. Our theorem is a quatitative analog of Furstenberg's Correspondence Principle.

We discuss selected topics of current research interest in the theory of dynamical systems, with emphasis on dimension theory, multifractal analysis, and quantitative recurrence. The topics include the quantitative versus the qualitative…