scholarly journals Skew products, quantitative recurrence, shrinking targets and decay of correlations

2014 ◽  
Vol 35 (6) ◽  
pp. 1814-1845 ◽  
Author(s):  
STEFANO GALATOLO ◽  
JÉRÔME ROUSSEAU ◽  
BENOIT SAUSSOL
Keyword(s):  
Dynamical Systems ◽  
Limit Distribution ◽  
Time Scale ◽  
Return Time ◽  
Decay Of Correlations ◽  
Polynomial Decay ◽  
Skew Products ◽  
Hyperbolic Dynamical Systems ◽  
Quantitative Recurrence ◽  
Scale Behavior

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 we show that those systems have a polynomial decay of correlations with respect to $C^{r}$ observables, and give estimations for its exponent, which depend on $r$ and on the arithmetical properties of the system. We also show examples of systems of this kind having no shrinking target property, and having a trivial limit distribution of return time statistics.

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

2012 ◽  
Vol 33 (2) ◽  
pp. 475-498 ◽  
Author(s):  
NICOLAI HAYDN ◽  
MATTHEW NICOL ◽  
TOMAS PERSSON ◽  
SANDRO VAIENTI
Keyword(s):  
Dynamical Systems ◽  
Return Time ◽  
Decay Of Correlations ◽  
Cantelli Lemma ◽  
Target Problem ◽  
Exponential Decay Of Correlations ◽  
Hyperbolic Dynamical Systems ◽  
Rate Of Decay ◽  
Dispersing Billiards ◽  
Lozi Maps

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.

Stable topological transitivity properties of ℝn-extensions of hyperbolic transformations

2011 ◽  
Vol 32 (4) ◽  
pp. 1435-1443 ◽  
Author(s):  
A. MOSS ◽  
C. P. WALKDEN
Keyword(s):  
Dynamical Systems ◽  
Skew Product ◽  
Dense Orbit ◽  
Anosov Diffeomorphism ◽  
Topological Transitivity ◽  
Axiom A ◽  
Skew Products ◽  
Hyperbolic Flow ◽  
Hyperbolic Dynamical Systems ◽  
Dense Set

AbstractWe consider ℝn skew-products of a class of hyperbolic dynamical systems. It was proved by Niţică and Pollicott [Transitivity of Euclidean extensions of Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 25 (2005), 257–269] that for an Anosov diffeomorphism ϕ of an infranilmanifold Λ there is (subject to avoiding natural obstructions) an open and dense set f:Λ→ℝN for which the skew-product ϕf(x,v)=(ϕ(x),v+f(x)) on Λ×ℝN has a dense orbit. We prove a similar result in the context of an Axiom A hyperbolic flow on an attractor.

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

2012 ◽  
Vol 33 (1) ◽  
pp. 49-80 ◽  
Author(s):  
J.-R. CHAZOTTES ◽  
P. COLLET
Keyword(s):  
Dynamical Systems ◽  
Poisson Approximation ◽  
Return Time ◽  
Time Function ◽  
Exponential Tail ◽  
Approximation Result ◽  
One Dimensional ◽  
Hyperbolic Dynamical Systems ◽  
Unstable Manifolds ◽  
Number Of Visits

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.

Statistics of closest return for some non-uniformly hyperbolic systems

2001 ◽  
Vol 21 (2) ◽  
pp. 401-420 ◽  
Author(s):  
P. COLLET
Keyword(s):  
Dynamical Systems ◽  
Exponential Decay ◽  
Reference Point ◽  
Hyperbolic Systems ◽  
Initial Point ◽  
Decay Of Correlations ◽  
Exponential Decay Of Correlations ◽  
Hyperbolic Dynamical Systems ◽  
Uniformly Hyperbolic Systems ◽  
The Law

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.

scholarly journals Almost all $S$-integer dynamical systems have many periodic points

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
Author(s):  
T. B. WARD
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Zeta Function ◽  
Periodic Points ◽  
Radius Of Convergence ◽  
Dynamical Zeta Function ◽  
Hyperbolic Dynamical Systems ◽  
Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

Sign in / Sign up

Export Citation Format

Share Document