75 research outputs found

### Recurrence and lyapunov exponents

We prove two inequalities between the Lyapunov exponents of a diffeomorphism
and its local recurrence properties. We give examples showing that each of the
inequalities is optimal

### Recurrence spectrum in smooth dynamical systems

We prove that for conformal expanding maps the return time does have constant
multifractal spectrum. This is the counterpart of the result by Feng and Wu in
the symbolic setting

### Entropy and Poincar\'e recurrence from a geometrical viewpoint

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. Moreover, we show that minimal return times to dynamical balls grow
linearly with respect to its length. Finally, some interesting relations
between recurrence, dimension, entropy and Lyapunov exponents of ergodic
measures are given.Comment: 11 pages, revised versio

### On the statistical distribution of first--return times of balls and cylinders in chaotic systems

We study returns in dynamical systems: when a set of points, initially
populating a prescribed region, swarms around phase space according to a
deterministic rule of motion, we say that the return of the set occurs at the
earliest moment when one of these points comes back to the original region. We
describe the statistical distribution of these "first--return times" in various
settings: when phase space is composed of sequences of symbols from a finite
alphabet (with application for instance to biological problems) and when phase
space is a one and a two-dimensional manifold. Specifically, we consider
Bernoulli shifts, expanding maps of the interval and linear automorphisms of
the two dimensional torus. We derive relations linking these statistics with
Renyi entropies and Lyapunov exponents.Comment: submitted to Int. J. Bifurcations and Chao

### Correlation decay and recurrence estimates for some robust nonuniformly hyperbolic maps

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 $V_\theta$
of functions with essential bounded variation that strictly contain Holder
continuous observables. We deduce that the equilibrium states have exponential
decay of correlations. Furthermore, we prove exponential asymptotic distribu-
tion of hitting times and log-normal fluctuations of the return times around
the average given by the metric entropy.Comment: 24 page

### Rare events, escape rates and quasistationarity: some exact formulae

We present a common framework to study decay and exchanges rates in a wide
class of dynamical systems. Several applications, ranging form the metric
theory of continuons fractions and the Shannon capacity of contrained systems
to the decay rate of metastable states, are given

### 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.Comment: To appear in Communications in Mathematical Physic

### Absolutely continuous invariant measures for random non-uniformly expanding maps

We prove existence of (at most denumerable many) absolutely continuous
invariant probability measures for random one-dimensional dynamical systems
with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is
bounded away from zero, we obtain finitely many ergodic absolutely continuous
invariant probability measures, describing the asymptotics of almost every
point.
We also prove a similar result for higher-dimensional random non-uniformly
expanding dynamical systems. The results are consequences of the construction
of such measures for skew-products with essentially arbitrary base dynamics and
asymptotic expansion along the fibers. In both cases our method deals with
either critical or singular points for the random maps.Comment: 30 pages; 2 figures. Keywords: non-uniform expansion, random
dynamics, slow recurrence, singular and critical set, absolutely continuous
invariant measures, skew-product. To appear in Math Z, 201

### Multiscale Systems, Homogenization, and Rough Paths:VAR75 2016: Probability and Analysis in Interacting Physical Systems

In recent years, substantial progress was made towards understanding
convergence of fast-slow deterministic systems to stochastic differential
equations. In contrast to more classical approaches, the assumptions on the
fast flow are very mild. We survey the origins of this theory and then revisit
and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1
(2016), 479-520], taking into account recent progress in $p$-variation and
c\`adl\`ag rough path theory.Comment: 27 pages. Minor corrections. To appear in Proceedings of the
Conference in Honor of the 75th Birthday of S.R.S. Varadha

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