Invariant' measures in hydrodynamic systems with random perturbations

2008 ◽  
pp. 107-113
Author(s):  
Ana Bela Cruzeiro
Keyword(s):  
Invariant Measures ◽  
Random Perturbations ◽  
Hydrodynamic Systems

scholarly journals Degenerate random perturbations of Anosov diffeomorphisms

2009 ◽  
Vol 30 (3) ◽  
pp. 931-951 ◽  
Author(s):  
TATIANA YARMOLA
Keyword(s):  
Riemannian Manifolds ◽  
Invariant Measures ◽  
Random Perturbations ◽  
Absolutely Continuous ◽  
Stable Manifolds ◽  
Anosov Diffeomorphisms ◽  
Absolutely Continuous Invariant Measures ◽  
Generic Rank ◽  
Toral Automorphisms ◽  
Absolutely Continuous Invariant

AbstractThis paper deals with random perturbations of diffeomorphisms on n-dimensional Riemannian manifolds with distributions supported on k-dimensional disks, where k<n. First we demonstrate general but not very intuitive conditions which guarantee that all invariant measures for rank-k random perturbations of C2 diffeomorphisms are absolutely continuous with respect to the Riemannian measure on M. For two subclasses of Anosov diffeomorphisms, hyperbolic toral automorphisms and Anosov diffeomorphisms with codimension 1 stable manifolds, the above conditions are modified in order to relate k-dimensional disks that support the distributions to certain foliations that arise from Anosov diffeomorphisms. We conclude that generic rank-k random perturbations have absolutely continuous invariant measures.

Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps

1992 ◽  
Vol 12 (1) ◽  
pp. 13-37 ◽  
Author(s):  
Michael Benedicks ◽  
Lai-Sang Young
Keyword(s):  
Positive Measure ◽  
Invariant Measures ◽  
Random Perturbations ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Small Random Perturbations ◽  
One Dimensional Maps ◽  
Absolutely Continuous Invariant

AbstractWe study the quadratic family and show that for a positive measure set of parameters the map has an absolutely continuous invariant measure that is stable under small random perturbations.

scholarly journals On the liftability of expanding stationary measures

2021 ◽  
pp. 2150039
Author(s):  
José F. Alves ◽  
Carla L. Dias ◽  
Helder Vilarinho
Keyword(s):  
Riemannian Manifold ◽  
Finite Number ◽  
Lyapunov Exponents ◽  
Invariant Measures ◽  
Stochastic Perturbation ◽  
Random Perturbations ◽  
Absolutely Continuous ◽  
Local Diffeomorphism ◽  
Stationary Measures ◽  
Topologically Transitive

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is a random Gibbs–Markov–Young structure which can be used to lift that measure. We also prove that if the original map admits a finite number of expanding invariant measures then the stationary measures of a sufficiently small stochastic perturbation are expanding.

Extending Quasi-Invariant Measures by Using Subgroups of a Given Group

2003 ◽  
Vol 10 (2) ◽  
pp. 247-255
Author(s):  
A. Kharazishvili
Keyword(s):  
Invariant Measures ◽  
Uncountable Group

Abstract A method of extending σ-finite quasi-invariant measures given on an uncountable group, by using a certain family of its subgroups, is investigated.

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

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