30 research outputs found
Recurrence rate in rapidly mixing dynamical systems
For measure preserving dynamical systems on metric spaces we study the time
needed by a typical orbit to return back close to its starting point. We prove
that when the decay of correlation is super-polynomial the recurrence rates and
the pointwise dimensions are equal. This gives a broad class of systems for
which the recurrence rate equals the Hausdorff dimension of the invariant
measure
Quantitative recurrence in two-dimensional extended processes
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 the case of some extended dynamical systems similar to planar
random walks, including \ZZ^2-extension of hyperbolic dynamics. We define a
pointwise recurrence rate and relate it to the dimension of the process, and
establish a convergence in distribution of the rescaled return times near the
origin
Central limit theorem for dimension of Gibbs measures for skew expanding maps
We consider a class of non-conformal expanding maps on the -dimensional
torus. For an equilibrium measure of an H\"older potential, we prove an
analogue of the Central Limit Theorem for the fluctuations of the logarithm of
the measure of balls as the radius goes to zero. An unexpected consequence is
that when the measure is not absolutely continuous, then half of the balls of
radius \eps have a measure smaller than \eps^\delta and half of them have a
measure larger than \eps^\delta, where is the Hausdorff dimension of
the measure. We first show that the problem is equivalent to the study of the
fluctuations of some Birkhoff sums. Then we use general results from
probability theory as the weak invariance principle and random change of time
to get our main theorem. Our method also applies to conformal repellers and
Axiom A surface diffeomorphisms and possibly to a class of one-dimensional non
uniformly expanding maps. These generalizations are presented at the end of the
paper
Back to balls in billiards
We consider a billiard in the plane with periodic configuration of convex
scatterers. This system is recurrent, in the sense that almost every orbit
comes back arbitrarily close to the initial point. In this paper we study the
time needed to get back in an r-ball about the initial point, in the phase
space and also for the position, in the limit when r->0. We establish the
existence of an almost sure convergence rate, and prove a convergence in
distribution for the rescaled return times
Large deviations for return times in non-rectangle sets for axiom A diffeomorphisms
For Axiom A diffeomorphisms and equilibrium states, we prove a Large
deviations result for the sequence of successive return times into a fixed
Borel set, under some assumption on the boundary. Our result relies on and
extends the work by Chazottes and Leplaideur who considered cylinder sets of a
Markov partition
Hitting and returning into rare events for all alpha-mixing processes
We prove that for any -mixing stationnary process the hitting time of
any -string converges, when suitably normalized, to an exponential
law. We identify the normalization constant . A similar statement
holds also for the return time. To establish this result we prove two other
results of independent interest. First, we show a relation between the rescaled
hitting time and the rescaled return time, generalizing a theorem by Haydn,
Lacroix and Vaienti. Second, we show that for positive entropy systems, the
probability of observing any -string in consecutive observations, goes
to zero as goes to infinity
Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing
We consider some nonuniformly 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 0 and after suitable normalization.Comment: 21 pages, 3 figure