scholarly journals On typical Markov operators acting on Borel measures

2005 ◽  
Vol 2005 (5) ◽  
pp. 489-497 ◽  
Author(s):  
Tomasz Szarek
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Markov Operator ◽  
Baire Category ◽  
Asymptotically Stable ◽  
Markov Operators ◽  
Borel Measures

It is proved that, in the sense of Baire category, almost every Markov operator acting on Borel measures is asymptotically stable and the Hausdorff dimension of its invariant measure is equal to zero.

scholarly journals Algebraic polymorphisms

2008 ◽  
Vol 28 (2) ◽  
pp. 633-642 ◽  
Author(s):  
KLAUS SCHMIDT ◽  
ANATOLY VERSHIK
Keyword(s):  
Algebraic Geometry ◽  
Invariant Measure ◽  
Special Class ◽  
Markov Operator ◽  
Point Of View ◽  
Compact Groups ◽  
Rational Matrix ◽  
Conjugate Operator ◽  
Markov Operators ◽  
Toral Automorphisms

AbstractIn this paper we consider a special class of polymorphisms with invariant measure, the algebraic polymorphisms of compact groups. A general polymorphism is—by definition—a many-valued map with invariant measure, and the conjugate operator of a polymorphism is a Markov operator (i.e. a positive operator on L2 of norm 1 which preserves the constants). In the algebraic case a polymorphism is a correspondence in the sense of algebraic geometry, but here we investigate it from a dynamical point of view. The most important examples are the algebraic polymorphisms of a torus, where we introduce a parametrization of the semigroup of toral polymorphisms in terms of rational matrices and describe the spectra of the corresponding Markov operators. A toral polymorphism is an automorphism of $\mathbb {T}^m$ if and only if the associated rational matrix lies in $\mathrm {GL}(m,\mathbb {Z})$. We characterize toral polymorphisms which are factors of toral automorphisms.

Factorization of an adjontable Markov operator

2019 ◽  
Vol 22 (02) ◽  
pp. 1950013
Author(s):  
Carlo Pandiscia
Keyword(s):  
Hilbert Space ◽  
Matrix Algebra ◽  
Unitary Operator ◽  
Markov Operator ◽  
Factorization Property ◽  
Markov Operators ◽  
Dilation Theory ◽  
Probability Spaces

In this work, we propose a method to investigate the factorization property of a adjontable Markov operator between two algebraic probability spaces without using the dilation theory. Assuming the existence of an anti-unitary operator on Hilbert space related to Stinespring representations of our Markov operator, which satisfy some particular modular relations, we prove that it admits a factorization. The method is tested on the two typologies of maps which we know admits a factorization, the Markov operators between commutative probability spaces and adjontable homomorphism. Subsequently, we apply these methods to particular adjontable Markov operator between matrix algebra which fixes the diagonal.

Bi-invariant sets and measures have integer Hausdorff dimension

1999 ◽  
Vol 19 (2) ◽  
pp. 523-534 ◽  
Author(s):  
DAVID MEIRI ◽  
YUVAL PERES
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Ergodic Measure ◽  
Invariant Set ◽  
Invariant Sets ◽  
Dimensional Torus ◽  
Uniformly Distributed Sequence ◽  
Invariant Ergodic Measure

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.

scholarly journals Weak Markov operators

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5453-5457
Author(s):  
Hūlya Duru ◽  
Serkan Ilter
Keyword(s):  
Positive Operator ◽  
Extreme Points ◽  
Markov Operator ◽  
Weak Order ◽  
Markov Operators

Let A and B be f -algebras with unit elements eA and eB respectively. A positive operator T from A to B satisfying T(eA) = eB is called a Markov operator. In this definition we replace unit elements with weak order units and, in this case, call T to be a weak Markov operator. In this paper, we characterize extreme points of the weak Markov operators.

scholarly journals A relation between Lyapunov exponents, Hausdorff dimension and entropy

1981 ◽  
Vol 1 (4) ◽  
pp. 451-459 ◽  
Author(s):  
Anthony Manning
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Unstable Manifold ◽  
Positive Lyapunov Exponent ◽  
Axiom A ◽  
Generic Points

AbstractFor an Axiom A diffeomorphism of a surface with an ergodic invariant measure we prove that the entropy is the product of the positive Lyapunov exponent and the Hausdorff dimension of the set of generic points in an unstable manifold.

scholarly journals Exponential Convergence For Markov Systems

2015 ◽  
Vol 29 (1) ◽  
pp. 139-149 ◽  
Author(s):  
Maciej Ślęczka
Keyword(s):  
Invariant Measure ◽  
Exponential Convergence ◽  
Iterated Function Systems ◽  
Markov Systems ◽  
Markov Operators ◽  
Iterated Function

AbstractMarkov operators arising from graph directed constructions of iterated function systems are considered. Exponential convergence to an invariant measure is proved.

The Hausdorff Dimension of an Ergodic Invariant Measure for a Piecewise Monotonic Map of the Interval

1992 ◽  
Vol 35 (1) ◽  
pp. 84-98 ◽  
Author(s):  
Franz Hofbauer ◽  
Peter Raith
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Bounded Variation ◽  
Continuous Map ◽  
Piecewise Monotonic ◽  
Piecewise Continuous

AbstractWe consider a piecewise monotonie and piecewise continuous map T on the interval. If T has a derivative of bounded variation, we show for an ergodic invariant measure μ with positive Ljapunov exponent λμ that the Hausdorff dimension of μ equals hμ / λμ.

Hausdorff dimension of the harmonic measure on trees

1998 ◽  
Vol 18 (3) ◽  
pp. 631-660 ◽  
Author(s):  
VADIM A. KAIMANOVICH
Keyword(s):  
Hausdorff Dimension ◽  
Random Walks ◽  
Large Class ◽  
Harmonic Measure ◽  
Random Environment ◽  
Free Groups ◽  
Equivalence Relations ◽  
Markov Operators ◽  
Rate Of Escape ◽  
Asymptotic Entropy

For a large class of Markov operators on trees we prove the formula ${\bf HD}\,\nu=h/l$ connecting the Hausdorff dimension of the harmonic measure $\nu$ on the tree boundary, the rate of escape $l$ and the asymptotic entropy $h$. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations.

A dimension formula for endomorphisms—the Belykh family

1998 ◽  
Vol 18 (5) ◽  
pp. 1283-1309 ◽  
Author(s):  
J. SCHMELING
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Parameter Family ◽  
Full Measure ◽  
Srb Measure ◽  
Dimension Formula

In [10] we considered a class of hyperbolic endomorphisms and asked the question whether there exists a physically motivated invariant measure (SRB measure) and if so we gave a criterion when the map is invertible on a set of full measure. In this work we want to consider a particular example of this class — in fact a three-parameter family of those — and prove that a.s. the criterion is fulfilled. From this it follows that the Young formulae for the Hausdorff dimension of the SRB measure holds.

Sign in / Sign up

Export Citation Format

Share Document