scholarly journals Polynomial decay of correlations for a class of smooth flows on the two torus

2001 ◽  
Vol 129 (4) ◽  
pp. 487-503 ◽  
Author(s):  
Bassam Fayad
Keyword(s):  
Decay Of Correlations ◽  
Polynomial Decay

scholarly journals Polynomial decay of correlations in linked-twist maps

2013 ◽  
Vol 34 (5) ◽  
pp. 1724-1746 ◽  
Author(s):  
J. SPRINGHAM ◽  
R. STURMAN
Keyword(s):  
Decay Of Correlations ◽  
Polynomial Decay ◽  
Arnold Cat Map ◽  
Twist Maps ◽  
Area Preserving

AbstractLinked-twist maps are area-preserving, piecewise diffeomorphisms, defined on a subset of the torus. They are non-uniformly hyperbolic generalizations of the well-known Arnold cat map. We show that a class of canonical examples have polynomial decay of correlations for$\alpha $-Hölder observables, of order$1/ n$.

scholarly journals POLYNOMIAL DECAY OF CORRELATIONS IN THE GENERALIZED BAKER'S TRANSFORMATION

2013 ◽  
Vol 23 (08) ◽  
pp. 1350130 ◽  
Author(s):  
CHRISTOPHER BOSE ◽  
RUA MURRAY
Keyword(s):  
Hyperbolic Systems ◽  
Decay Of Correlations ◽  
Polynomial Decay ◽  
Mixing Rate ◽  
New Techniques ◽  
Correlation Decay ◽  
Simple Geometry ◽  
Uniformly Hyperbolic Systems ◽  
Mixing Rates ◽  
Area Preserving

We introduce a family of area preserving generalized baker's transformations acting on the unit square and having sharp polynomial rates of mixing for Hölder data. The construction is geometric, relying on the graph of a single variable "cut function". Each baker's map B is nonuniformly hyperbolic and while the exact mixing rate depends on B, all polynomial rates can be attained. The analysis of mixing rates depends on building a suitable Young tower for an expanding factor. The mechanisms leading to a slow rate of correlation decay are especially transparent in our examples due to the simple geometry in the construction. For this reason, we propose this class of maps as an excellent testing ground for new techniques for the analysis of decay of correlations in non-uniformly hyperbolic systems. Finally, some of our examples can be seen to be extensions of certain 1D non-uniformly expanding maps that have appeared in the literature over the last twenty years, thereby providing a unified treatment of these interesting and well-studied examples.

Polynomial decay of correlations for almost Anosov diffeomorphisms

2017 ◽  
Vol 39 (3) ◽  
pp. 832-864
Author(s):  
XU ZHANG ◽  
HUYI HU
Keyword(s):  
Fixed Point ◽  
Decay Of Correlations ◽  
Polynomial Decay ◽  
Lower And Upper Bounds ◽  
Third Order ◽  
Anosov Diffeomorphisms ◽  
The Third ◽  
Srb Measures ◽  
Taylor Expansions ◽  
Almost Anosov

We investigate the polynomial lower and upper bounds for decay of correlations of a class of two-dimensional almost Anosov diffeomorphisms with respect to their Sinai–Ruelle–Bowen (SRB) measures. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the indifferent fixed point, and are expressed by using coefficients of the third-order terms in the Taylor expansions of the diffeomorphisms at the indifferent fixed point.

scholarly journals Polynomial Decay of Correlations for Flows, Including Lorentz Gas Examples

2019 ◽  
Vol 368 (1) ◽  
pp. 55-111 ◽  
Author(s):  
Péter Bálint ◽  
Oliver Butterley ◽  
Ian Melbourne
Keyword(s):  
Lorentz Gas ◽  
Decay Of Correlations ◽  
Polynomial Decay

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 Lower bounds for the decay of correlations in non-uniformly expanding maps

2017 ◽  
Vol 39 (7) ◽  
pp. 1936-1970 ◽  
Author(s):  
HUYI HU ◽  
SANDRO VAIENTI
Keyword(s):  
Lower Bounds ◽  
Transfer Operator ◽  
Type Inequality ◽  
Decay Of Correlations ◽  
Polynomial Decay ◽  
Expanding Maps ◽  
Polynomial Type ◽  
Higher Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Markov Structure

We give conditions under which non-uniformly 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 is satisfied in a Banach space ${\mathcal{B}}$, 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 ${\mathcal{B}}.$ We also provide some general conditions that give aperiodicity for expanding maps in higher dimensional spaces. As applications, we obtain lower bounds for piecewise expanding maps with an indifferent fixed point and for which we also allow non-Markov structure and unbounded distortion. The observables are functions that have bounded variation or satisfy quasi-Hölder conditions and have their support bounded away from the neutral fixed points.

scholarly journals Mixing rates for potentials of non-summable variations

2021 ◽  
pp. 1-18
Author(s):  
CHRISTOPHE GALLESCO ◽  
DANIEL Y. TAKAHASHI
Keyword(s):  
Invariance Principle ◽  
Type Inequality ◽  
Upper Bounds ◽  
Decay Of Correlations ◽  
Relaxation Rates ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
Coupling Inequality ◽  
Mixing Rates

Abstract Mixing rates, relaxation rates, and decay of correlations for dynamics defined by potentials with summable variations are well understood, but little is known for non-summable variations. This paper exhibits upper bounds for these quantities for dynamics defined by potentials with square-summable variations. We obtain these bounds as corollaries of a new block coupling inequality between pairs of dynamics starting with different histories. As applications of our results, we prove a new weak invariance principle and a Hoeffding-type inequality.

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