SOBOLEV REGULARITY OF INVARIANT MEASURES FOR GENERALIZED ORNSTEIN–UHLENBECK OPERATORS

2006 ◽  
Vol 09 (04) ◽  
pp. 595-606 ◽  
Author(s):  
ABDELHADI ES-SARHIR
Keyword(s):  
Hilbert Space ◽  
Invariant Measure ◽  
Sobolev Space ◽  
Invariant Measures ◽  
Separable Hilbert Space ◽  
Square Root ◽  
Absolutely Continuous ◽  
Sobolev Regularity ◽  
Reference Measure ◽  
Nikodym Derivative

This paper deals with the regularity of an invariant measure μ associated to a class of generalized Ornstein–Uhlenbeck operators. Regularity here means that μ is absolutely continuous with respect to a properly chosen Gaussian reference measure σ on a separable Hilbert space H. Moreover, the square root of its Radon–Nikodym derivative ρ should belong to some directional Sobolev space [Formula: see text].

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.

CONVERGENCE OF INVARIANT MEASURES FOR DEGENERATING SEQUENCES OF HYPERBOLIC MAPPINGS WITH SINGULARITIES

1996 ◽  
Vol 06 (06) ◽  
pp. 1143-1151
Author(s):  
E. A. SATAEV
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant ◽  
Expanding Mapping

This paper is devoted to presenting and giving a sketch of the proof of the theorem which states that, if the sequence of hyperbolic mappings with singularities converges to degenerating piecewise expanding mapping, then the corresponding sequence of measures of a Sinai-Bowen-Ruelle type converges to an absolutely continuous invariant measure.

scholarly journals Decay of correlations in suspension semi-flows of angle-multiplying maps

2008 ◽  
Vol 28 (1) ◽  
pp. 291-317 ◽  
Author(s):  
MASATO TSUJII
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Spectral Radius ◽  
Decay Of Correlations ◽  
Square Root ◽  
Precise Description ◽  
Anisotropic Sobolev Space ◽  
Essential Spectral Radius ◽  
Frobenius Operator ◽  
Dense Set

AbstractWe consider suspension semi-flows of angle-multiplying maps on the circle for Cr ceiling functions with r≥3. Under a Crgeneric condition on the ceiling function, we show that there exists a Hilbert space (anisotropic Sobolev space) contained in the L2 space such that the Perron–Frobenius operator for the time-t-map acts naturally on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum expansion rate. This leads to a precise description of decay of correlations. Furthermore, the Perron–Frobenius operator for the time-t-map is quasi-compact for a Cr open and dense set of ceiling functions.

scholarly journals Estimating invariant measures and Lyapunov exponents

1996 ◽  
Vol 16 (4) ◽  
pp. 735-749 ◽  
Author(s):  
Brian R. Hunt
Keyword(s):  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Recent Interest ◽  
Absolutely Continuous Invariant Measure ◽  
Time Averages ◽  
Absolutely Continuous Invariant

AbstractThis paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius—Perron operator for such maps. The method is applied to estimate the Lyapunov exponents for a planar map of recent interest.

scholarly journals Absolutely continuous invariant measures for non-uniformly expanding maps

2009 ◽  
Vol 29 (4) ◽  
pp. 1185-1215 ◽  
Author(s):  
HUYI HU ◽  
SANDRO VAIENTI
Keyword(s):  
Invariant Measure ◽  
Fixed Points ◽  
Large Class ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractFor a large class of non-uniformly expanding maps of ℝm, with indifferent fixed points and unbounded distortion and that are non-necessarily Markovian, we construct an absolutely continuous invariant measure. We extend previously used techniques for expanding maps on quasi-Hölder spaces to our case. We give general conditions and provide examples to which our results apply.

Absolutely continuous invariant measures for piecewise expanding C2 transformations in Rn on domains with cusps on the boundaries

1996 ◽  
Vol 16 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Kourosh Adl-Zarabi
Keyword(s):  
Lower Bound ◽  
Invariant Measure ◽  
Finite Number ◽  
Invariant Measures ◽  
Bounded Region ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Derivative Matrix ◽  
Absolutely Continuous Invariant

AbstractLet Ω be a bounded region in Rn and let be a partition of Ω into a finite number of subsets having piecewise C2 boundaries. The boundaries may contain cusps. Let τ: Ω → Ω be piecewise C2 on and expanding in the sense that there exists α > 1 such that for any i = 1, 2,…,m, where is the derivative matrix of and ‖·‖ is the euclidean matrix norm. The main result provides a lower bound on α which guarantees the existence of an absolutely continuous invariant measure for τ.

scholarly journals Absolutely continuous invariant measures for some piecewise hyperbolic affine maps

2008 ◽  
Vol 28 (1) ◽  
pp. 211-228 ◽  
Author(s):  
TOMAS PERSSON
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Bounded Subset ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Piecewise Affine ◽  
Affine Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractA class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.

scholarly journals Maximal absolutely continuous invariant measures for piecewise linear Markov transformations

1990 ◽  
Vol 10 (4) ◽  
pp. 645-656 ◽  
Author(s):  
W. Byers ◽  
P. Góra ◽  
A. Boyarsky
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Piecewise Linear ◽  
Maximal Entropy ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant Measure ◽  
Necessary And Sufficient ◽  
Absolutely Continuous Invariant

AbstractLet be an irreducible 0–1 matrix such that the non-zero entries in each row are consecutive. Let be the class of piecewise linear Markov transformations τ on [0, 1] into [0, 1] induced by for which the absolutely continuous invariant measure has maximal entropy. The main result presents necessary and sufficient slope conditions on τ which guarantee that τ ∈ .

scholarly journals Almost everywhere convergence of ergodic series

2015 ◽  
Vol 37 (2) ◽  
pp. 490-511 ◽  
Author(s):  
AIHUA FAN
Keyword(s):  
Invariant Measure ◽  
Integrable Function ◽  
Gibbs Measures ◽  
Type Inequality ◽  
Convergent Series ◽  
Mean Value ◽  
Absolutely Continuous ◽  
Hyperbolic Dynamics ◽  
Almost Everywhere ◽  
Nikodym Derivative

We consider ergodic series of the form $\sum _{n=0}^{\infty }a_{n}f(T^{n}x)$, where $f$ is an integrable function with zero mean value with respect to a $T$-invariant measure $\unicode[STIX]{x1D707}$. Under certain conditions on the dynamical system $T$, the invariant measure $\unicode[STIX]{x1D707}$ and the function $f$, we prove that the series converges $\unicode[STIX]{x1D707}$-almost everywhere if and only if $\sum _{n=0}^{\infty }|a_{n}|^{2}<\infty$, and that in this case the sum of the convergent series is exponentially integrable and satisfies a Khintchine-type inequality. We also prove that the system $\{f\circ T^{n}\}$ is a Riesz system if and only if the spectral measure of $f$ is absolutely continuous with respect to the Lebesgue measure and the Radon–Nikodym derivative is bounded from above as well as from below by a constant. We check the conditions for Gibbs measures $\unicode[STIX]{x1D707}$ relative to hyperbolic dynamics $T$ and for Hölder functions $f$. An application is given to the study of differentiability of the Weierstrass-type functions $\sum _{n=0}^{\infty }a_{n}f(3^{n}x)$.

CONSTRUCTING INVARIANT MEASURES FROM DATA

1995 ◽  
Vol 05 (04) ◽  
pp. 1181-1192 ◽  
Author(s):  
GARY FROYLAND ◽  
KEVIN JUDD ◽  
ALISTAIR I. MEES ◽  
DAVID WATSON ◽  
KENJI MURAO
Keyword(s):  
Dynamical System ◽  
Experimental Data ◽  
Phase Space ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Reconstruction Technique ◽  
Continuous Approximation ◽  
Absolutely Continuous ◽  
Finite Set

We present a method of approximating an invariant measure of a dynamical system from a finite set of experimental data. Our reconstruction technique automatically provides us with a partition of phase space, and we assign each set in the partition a certain weight. By refining the partition, we may make our approximation to an invariant measure of the reconstructed system as accurate as we wish. Our method provides us with both a singular and an absolutely continuous approximation, so that the most suitable representation may be chosen for a particular problem.

Sign in / Sign up

Export Citation Format

Share Document