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1
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A method of construction of an invariant measure

100%
Dawidowicz A.
Annales Polonici Mathematici
|
1992
|
tom 57
|
nr 3
205-208
EN
A method of construction of an invariant measure on a function space is presented.
2
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On the generalized Avez method

100%
Dawidowicz A.
Annales Polonici Mathematici
|
1992
|
tom 57
|
nr 3
209-218
EN
A generalization of the Avez method of construction of an invariant measure is presented.
3
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A version of non-Hamiltonian Liouville equation

100%
Rom C.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2014
|
tom 34
|
nr 1
5-14
EN
In this paper we give a version of the theorem on local integral invariants of systems of ordinary differential equations. We give, as an immediate conclusion of this theorem, a condition which guarantees existence of an invariant measure of local dynamical systems. Results of this type lead to the Liouville equation and have been frequently proved under various assumptions. Our method of the proof is simpler and more direct.
4
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Exponential Convergence For Markov Systems

100%
Ślęczka M.
Annales Mathematicae Silesianae
|
2015
|
tom 29
|
nr 1
139-149
EN
Markov operators arising from graph directed constructions of iterated function systems are considered. Exponential convergence to an invariant measure is proved.
5
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Invariant measures for iterated function systems

80%
Szarek T.
Annales Polonici Mathematici
|
2000
|
tom 75
|
nr 1
87-98
EN
A new criterion for the existence of an invariant distribution for Markov operators is presented. Moreover, it is also shown that the unique invariant distribution of an iterated function system is singular with respect to the Hausdorff measure.
6
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Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in $ℝ^d$

80%
Bugiel P.
Annales Polonici Mathematici
|
1998
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tom 68
|
nr 2
125-157
EN
Asymptotic properties of the sequences (a) ${P^j_φ g}_{j=1}^{∞}$ and (b) ${j^{-1} ∑_{i=0}^{j-1} Pⁱ_φ g}_{j=1}^{∞}$, where $P_φ:L¹ → L¹$ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = {f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1}. An operator-theoretic analogue of Rényi's Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov maps in $ℝ^d$. Also the Bernoulli property is proved for a class of smooth Markov maps in $ℝ^d$.
7
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On a nonstandard approach to invariant measures for Markov operators

80%
Wiśnicki A.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
|
2010
|
tom 64
|
nr 2
EN
We show the existence of invariant measures for Markov-Feller operators defined on completely regular topological spaces which satisfy the classical positivity condition.
8
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The uniqueness of Haar measure and set theory

80%
Zakrzewski P.
Colloquium Mathematicum
|
1997
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tom 74
|
nr 1
109-121
EN
Let G be a group of homeomorphisms of a nondiscrete, locally compact, σ-compact topological space X and suppose that a Haar measure on X exists: a regular Borel measure μ, positive on nonempty open sets, finite on compact sets and invariant under the homeomorphisms from G. Under some mild assumptions on G and X we prove that the measure completion of μ is the unique, up to a constant factor, nonzero, σ-finite, G-invariant measure defined on its domain iff μ is ergodic and the G-orbits of all points of X are uncountable. In particular, this is true if either G is a locally compact, σ-compact topological group acting continuously on X, or the space X is uniform and nonseparable, and G consists of uniformly equicontinuous unimorphisms of X.
9
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Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows

80%
Downarowicz T., Lacroix
Studia Mathematica
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1998
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tom 130
|
nr 2
149-170
EN
Let $(Z,T_Z)$ be a minimal non-periodic flow which is either symbolic or strictly ergodic. Any topological extension of $(Z,T_Z)$ is Borel isomorphic to an almost 1-1 extension of $(Z,T_Z)$. Moreover, this isomorphism preserves the affine-topological structure of the invariant measures. The above extends a theorem of Furstenberg-Weiss (1989). As an application we prove that any measure-preserving transformation which admits infinitely many rational eigenvalues is measure-theoretically isomorphic to a strictly ergodic toeplitz flow.
10
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Ergodic properties of skew products withfibre maps of Lasota-Yorke type

80%
Kowalski Z. S.
Applicationes Mathematicae
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1993-1995
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tom 22
|
nr 2
155-163
EN
We consider the skew product transformation T(x,y)= (f(x), $T_{e(x)}$) where f is an endomorphism of a Lebesgue space (X,A,p), e : X → S and ${T_s}_{s \in S}$ is a family of Lasota-Yorke type maps of the unit interval into itself. We obtain conditions under which the ergodic properties of f imply the same properties for T. Consequently, we get the asymptotical stability of random perturbations of a single Lasota-Yorke type map. We apply this to some probabilistic model of the motion of cogged bits in the rotary drilling of hard rock with high rotational speed.
11
Content available remote

Density of periodic orbit measures for transformations on the interval with two monotonic pieces

80%
Hofbauer F., Raith P.
Fundamenta Mathematicae
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1998
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tom 157
|
nr 2-3
221-234
EN
Transformations T:[0,1] → [0,1] with two monotonic pieces are considered. Under the assumption that T is topologically transitive and $h_{top}(T) > 0$, it is proved that the invariant measures concentrated on periodic orbits are dense in the set of all invariant probability measures.
12
Content available remote

On nearly selfoptimizing strategies for multiarmed bandit problems with controlled arms

70%
Drabik E.
Applicationes Mathematicae
|
1995-1996
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tom 23
|
nr 4
449-473
EN
Two kinds of strategies for a multiarmed Markov bandit problem with controlled arms are considered: a strategy with forcing and a strategy with randomization. The choice of arm and control function in both cases is based on the current value of the average cost per unit time functional. Some simulation results are also presented.
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