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 Explicit coupling argument for non-uniformly hyperbolic transformations

2018 ◽  
Vol 149 (1) ◽  
pp. 101-130 ◽  
Author(s):  
A. Korepanov ◽  
Z. Kosloff ◽  
I. Melbourne
Keyword(s):  
Spectral Properties ◽  
Transfer Operator ◽  
Decay Of Correlations ◽  
Exponential Polynomial ◽  
Expanding Maps ◽  
Stretched Exponential ◽  
Expanding Map ◽  
Mixing Rates ◽  
Rates Of Decay

The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. We verify that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map. For non-uniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated with the inducing time. Finally, for non-uniformly hyperbolic transformations, we obtain the corresponding estimates for rates of decay of correlations.

scholarly journals An optimal transportation approach to the decay of correlations for non-uniformly expanding maps

2018 ◽  
Vol 40 (3) ◽  
pp. 714-750 ◽  
Author(s):  
BENOÎT R. KLOECKNER
Keyword(s):  
Functional Analysis ◽  
Spectral Gap ◽  
Transfer Operator ◽  
Specific Property ◽  
Optimal Transportation ◽  
Decay Of Correlations ◽  
Expanding Maps ◽  
Uniform Topology ◽  
Frobenius Theorem ◽  
Constant Potential

We consider the transfer operators of non-uniformly expanding maps for potentials of various regularity, and show that a specific property of potentials (‘flatness’) implies a Ruelle–Perron–Frobenius theorem and a decay of the transfer operator of the same speed as that entailed by the constant potential. The method relies neither on Markov partitions nor on inducing, but on functional analysis and duality, through the simplest principles of optimal transportation. As an application, we notably show that for any map of the circle which is expanding outside an arbitrarily flat neutral point, the set of Hölder potentials exhibiting a spectral gap is dense in the uniform topology. The method applies in a variety of situations, including Pomeau–Manneville maps with regular enough potentials, or uniformly expanding maps of low regularity with their natural potential; we also recover in a united fashion variants of several previous results.

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.

CHAOTIC BEHAVIOR OF HIGHER DIMENSIONAL TRANSFORMATIONS DEFINED ON COUNTABLE PARTITIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 1045-1049
Author(s):  
A. BOYARSKY ◽  
Y. S. LOU
Keyword(s):  
Invariant Measure ◽  
Additional Condition ◽  
Invariant Measures ◽  
Chaotic Behavior ◽  
Finite Collection ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Higher Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

Jablonski maps are higher dimensional maps defined on rectangular partitions with each component a function of only one variable. It is well known that expanding Jablonski maps have absolutely continuous invariant measures. In this note we consider Jablonski maps defined on countable partitions. Such maps occur, for example, in multivariable number theoretic problems. The main result establishes the existence of an absolutely continuous invariant measure for Jablonski maps on a countable partition with the additional condition that the images of all the partition elements form a finite collection. An example is given.

On the Inner Radius of a Nodal Domain

2008 ◽  
Vol 51 (2) ◽  
pp. 249-260 ◽  
Author(s):  
Dan Mangoubi
Keyword(s):  
Riemannian Manifold ◽  
Lower Bounds ◽  
Type Inequality ◽  
Large Eigenvalue ◽  
Upper And Lower Bounds ◽  
Local Behavior ◽  
Nodal Domain ◽  
Closed Riemannian Manifold ◽  
Inner Radius

AbstractLet M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue λ. We give upper and lower bounds on the inner radius of the type C/λα(log λ)β. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincaré type inequality proved by Maz’ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.

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$1/ n$.

TIME LOWER BOUNDS FOR PERMUTATION ROUTING ON MULTI-DIMENSIONAL BUSED MESHES

1993 ◽  
Vol 03 (02) ◽  
pp. 129-138
Author(s):  
STEVEN CHEUNG ◽  
FRANCIS C.M. LAU
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Routing Problem ◽  
Higher Dimensional ◽  
Low Dimensional ◽  
Permutation Routing

We present time lower bounds for the permutation routing problem on three- and higher-dimensional n x…x n meshes with buses. We prove an (r–1)n/r lower bound for the general case of an r-dimensional bused mesh, r≥2, which is not as strong for low-dimensional as for higher-dimensional cases. We then use a different approach to construct a 0.705n lower bound for the three-dimensional case.

Sign in / Sign up

Export Citation Format

Share Document