scholarly journals Ergodic measures for the irrational rotation on the circle

1988 ◽  
Vol 45 (1) ◽  
pp. 133-141 ◽  
Author(s):  
William Moran
Keyword(s):  
Invariant Measures ◽  
Irrational Rotation ◽  
Suitable Choice ◽  
Ergodic Measures ◽  
Riesz Products ◽  
Nikodym Derivative

AbstractRiesz products are employed to give a construction of quasi-invariant ergodic measures under the irrational rotation of T. By suitable choice of the parameters such measures may be required to have Fourier-Stieltjes coefficients vanishing at infinity. We show further that these are the unique quasi-invariant measures on T with their associated Radon-Nikodym derivative.

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.

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].

scholarly journals Invariant measures for higher-rank hyperbolic abelian actions

1996 ◽  
Vol 16 (4) ◽  
pp. 751-778 ◽  
Author(s):  
A. Katok ◽  
R. J. Spatzier
Keyword(s):  
Invariant Measures ◽  
Metric Entropy ◽  
Ergodic Measures ◽  
Anosov Actions ◽  
Higher Rank ◽  
Partially Hyperbolic

AbstractWe investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of ℝk, ℤkandWe show that they are either Haar measures or that every element of the action has zero metric entropy.

scholarly journals On density of ergodic measures and generic points

2016 ◽  
Vol 38 (5) ◽  
pp. 1745-1767 ◽  
Author(s):  
KATRIN GELFERT ◽  
DOMINIK KWIETNIAK
Keyword(s):  
Invariant Measures ◽  
Metric Spaces ◽  
Probability Measures ◽  
Geodesic Flows ◽  
Generic Point ◽  
Negatively Curved ◽  
Ergodic Measures ◽  
Compact Case ◽  
Coded Systems ◽  
Generic Points

We introduce two properties of dynamical systems on Polish metric spaces: closeability and linkability. We show that they imply density of ergodic measures in the space of invariant probability measures and the existence of a generic point for every invariant measure. In the compact case, it follows from our conditions that the set of invariant measures is either a singleton of a measure concentrated on a periodic orbit or the Poulsen simplex. We provide examples showing that closability and linkability are independent properties. Our theory applies to systems with the periodic specification property, irreducible Markov chains over a countable alphabet, certain coded systems including $\unicode[STIX]{x1D6FD}$-shifts and $S$-gap shifts, $C^{1}$-generic diffeomorphisms of a compact manifold $M$ and certain geodesic flows of a complete connected negatively curved manifold.

scholarly journals Graph covers and ergodicity for zero-dimensional systems

2014 ◽  
Vol 36 (2) ◽  
pp. 608-631 ◽  
Author(s):  
TAKASHI SHIMOMURA
Keyword(s):  
Linear Dependence ◽  
Invariant Measures ◽  
Unique Ergodicity ◽  
Dimensional System ◽  
General Graph ◽  
Bratteli Diagrams ◽  
Linearly Independent ◽  
Ergodic Measures ◽  
Time Average ◽  
Path Number

Bratteli–Vershik systems have been widely studied. In the context of general zero-dimensional systems, Bratteli–Vershik systems are homeomorphisms that have Kakutani–Rohlin refinements. Bratteli diagrams are well suited to analyzing such systems. Besides this approach, general graph covers can be used to represent any zero-dimensional system. Indeed, all zero-dimensional systems can be described as certain kinds of sequences of graph covers that may not be brought about by Kakutani–Rohlin partitions. In this paper, we follow the context of general graph covers to analyze the relations between ergodic measures and circuits of graph covers. First, we formalize the condition for a sequence of graph covers to represent minimal Cantor systems. In constructing invariant measures, we deal with general compact metrizable zero-dimensional systems. In the context of Bratteli diagrams with finite rank, it has previously been mentioned that all ergodic measures should be limits of some combinations of towers of Kakutani–Rohlin refinements. We demonstrate this for the general zero-dimensional case, and develop a theorem that expresses the coincidence of the time average and the space average for ergodic measures. Additionally, we formulate a theorem that signifies the old relation between uniform convergence and unique ergodicity in the context of graph circuits for general zero-dimensional systems. Unlike previous studies, in our case of general graph covers there arises the possibility of the linear dependence of circuits. We give a condition for a full circuit system to be linearly independent. Previous research also showed that the bounded combinatorics imply unique ergodicity. We present a lemma that enables us to consider unbounded ranks of winding matrices. Finally, we present examples that are linked with a set of simple Bratteli diagrams having the equal path number property.

scholarly journals Invariant measures on the space of horofunctions of a word hyperbolic group

2009 ◽  
Vol 30 (1) ◽  
pp. 97-129 ◽  
Author(s):  
LEWIS BOWEN
Keyword(s):  
Equivalence Relation ◽  
Invariant Measures ◽  
Hyperbolic Group ◽  
Discrete Analog ◽  
Word Hyperbolic Group ◽  
Natural Equivalence ◽  
Ergodic Measures ◽  
Finite Space ◽  
Special Case ◽  
Measure Preserving Transformations

AbstractWe introduce a natural equivalence relation on the space ℋ0 of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen–Marcus theorem. Furthermore, if η is such a measure and G acts on a probability space (X,μ) by measure-preserving transformations then η×μ is virtually ergodic with respect to a natural equivalence relation on ℋ0×X. This is comparable to a special case of the Howe–Moore theorem. These results are applied to prove a new ergodic theorem for spherical averages in the case of a word hyperbolic group acting on a finite space.

Family of measures on a space of curves that are quasi-invariant with respect to some action of diffeomorphisms group

2016 ◽  
Vol 19 (03) ◽  
pp. 1650019
Author(s):  
Evgenii Dmitrievich Romanov
Keyword(s):  
Invariant Measures ◽  
Functional Space ◽  
Geometric Interpretation ◽  
Unitary Representations ◽  
Multidimensional Case ◽  
Dimensional Euclidean Space ◽  
Future Research ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations ◽  
Nikodym Derivative

A family of quasi-invariant measures on the special functional space of curves in a finite-dimensional Euclidean space with respect to the action of diffeomorphisms is constructed. The main result is an explicit expression for the Radon–Nikodym derivative of the transformed measure relative to the original one. The stochastic Ito integral allows to express the result in an invariant form for a wider class of diffeomorphisms. These measures can be used to obtain irreducible unitary representations of the diffeomorphisms group which will be studied in future research. A geometric interpretation of the action considered together with a generalization to the multidimensional case makes such representations applicable to problems of quantum mechanics.

scholarly journals Properties of invariant measures in dynamical systems with the shadowing property

2017 ◽  
Vol 38 (6) ◽  
pp. 2257-2294 ◽  
Author(s):  
JIAN LI ◽  
PIOTR OPROCHA
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Entropy Function ◽  
Shadowing Property ◽  
Topologically Transitive ◽  
Ergodic Measures ◽  
One To One ◽  
Transitive System

For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost one-to-one extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every $c\geq 0$ and $\unicode[STIX]{x1D700}>0$ the collection of ergodic measures (supported on almost one-to-one extensions of odometers) with entropy between $c$ and $c+\unicode[STIX]{x1D700}$ is dense in the space of invariant measures with entropy at least $c$. Moreover, if in addition the entropy function is upper semi-continuous, then, for every $c\geq 0$, ergodic measures with entropy $c$ are generic in the space of invariant measures with entropy at least $c$.

scholarly journals On conjugations of circle homeomorphisms with two break points

2012 ◽  
Vol 34 (3) ◽  
pp. 725-741 ◽  
Author(s):  
HABIBULLA AKHADKULOV ◽  
AKHTAM DZHALILOV ◽  
DIETER MAYER
Keyword(s):  
Lebesgue Measure ◽  
Rotation Number ◽  
Invariant Measures ◽  
Irrational Rotation ◽  
Singular Function ◽  
Almost Everywhere ◽  
Break Points ◽  
Irrational Rotation Number

AbstractLetfi∈C2+α(S1∖{ai,bi}),α>0,i=1,2, be circle homeomorphisms with two break pointsai,bi, that is, discontinuities in the derivativeDfi, with identical irrational rotation numberρandμ1([a1,b1])=μ2([a2,b2]), whereμiare the invariant measures offi,i=1,2. Suppose that the products of the jump ratios ofDf1andDf2do not coincide, that is,Df1(a1−0)/Df1(a1+0)⋅Df1(b1−0)/Df1(b1+0)≠Df2(a2−0)/Df2(a2+0)⋅Df2(b2−0)/Df2(b2+0) . Then the mapψconjugatingf1andf2is a singular function, that is, it is continuous onS1, butDψ(x)=0 almost everywhere with respect to Lebesgue measure.

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