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 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 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 On Gibbs Measures and Spectra of Ruelle Transfer Operators

2017 ◽  
Vol 60 (2) ◽  
pp. 411-421
Author(s):  
Luchezar Stoyanov
Keyword(s):  
Spectral Radius ◽  
Spectral Properties ◽  
Gibbs Measures ◽  
Transfer Operator ◽  
Frobenius Theorem ◽  
Transfer Operators

AbstractWe prove a comprehensive version of the Ruelle–Perron–Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Hölder constant of the function generating the operator appears only polynomially, not exponentially as in previously known estimates.

scholarly journals Effective high-temperature estimates for intermittent maps

2017 ◽  
Vol 39 (8) ◽  
pp. 2159-2175
Author(s):  
BENOÎT R. KLOECKNER
Keyword(s):  
Perturbation Theory ◽  
High Temperature ◽  
Limit Theorem ◽  
Spectral Gap ◽  
Lipschitz Constant ◽  
Transfer Operator ◽  
Linear Operators ◽  
Suitable Assumption ◽  
Unimodal Map ◽  
Definition Of

Using quantitative perturbation theory for linear operators, we prove a spectral gap for transfer operators of various families of intermittent maps with almost constant potentials (‘high-temperature’ regime). Hölder and bounded $p$-variation potentials are treated, in each case under a suitable assumption on the map, but the method should apply more generally. It is notably proved that for any Pommeau–Manneville map, any potential with Lipschitz constant less than 0.0014 has a transfer operator acting on $\operatorname{Lip}([0,1])$ with a spectral gap; and that for any two-to-one unimodal map, any potential with total variation less than 0.0069 has a transfer operator acting on $\operatorname{BV}([0,1])$ with a spectral gap. We also prove under quite general hypotheses that the classical definition of spectral gap coincides with the formally stronger one used in Giulietti et al [The calculus of thermodynamical formalism. J. Eur. Math. Soc., to appear. Preprint, 2015, arXiv:1508.01297], allowing all results there to be applied under the high-temperature bounds proved here: analyticity of pressure and equilibrium states, central limit theorem, etc.

scholarly journals Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

2021 ◽  
pp. 1-37
Author(s):  
WAEL BAHSOUN ◽  
CARLANGELO LIVERANI
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Absolute Continuity ◽  
Hyperbolic Systems ◽  
Transfer Operator ◽  
Statistical Properties ◽  
Functions Of Bounded Variation ◽  
Expanding Maps ◽  
New Information ◽  
Piecewise Expanding Maps

Abstract Given any smooth Anosov map, we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that, in the case of expanding maps, it reduces exactly to the usual space of functions of bounded variation which has proved to be particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations, which provides new information that could prove useful in treating hyperbolic systems with singularities.

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