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1
Content available remote 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.
2
Content available remote 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.
3
Content available remote 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.
4
Content available 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.
5
Content available remote The uniqueness of Haar measure and set theory
80%
Zakrzewski P.
Colloquium Mathematicum
|
1997
|
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.
6
Content available remote 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
|
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
Content available remote Random Iteration with Place Dependent Probabilities
80%
Kapica R. , Ślęczka M.
Probability and Mathematical Statistics
|
2020
|
tom Vol. 40, Fasc. 1
119--137
EN
We consider Markov chains arising from random iteration of functions Sθ : X → X, θ ϵ Θ, where X is a Polish space and Θ is an arbitrary set of indices. At x ϵ X, θ is sampled from a distribution ϑx on Θ, and the ϑx are different for different x. Exponential convergence to a unique invariant measure is proved. This result is applied to the case of random affine transformations on Rd, giving the existence of exponentially attractive perpetuities with place dependent probabilities.
8
Content available remote Density of periodic orbit measures for transformations on the interval with two monotonic pieces
80%
Hofbauer F. , Raith P.
Fundamenta Mathematicae
|
1998
|
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.
9
Some results of ergodic theory and their application to drilling modeling
80%
Chakvetadze G.
Prace Naukowe Instytutu Matematyki Politechniki Wrocławskiej. Konferencje
|
2001
|
tom Vol. 24, nr 3
7-16
10
Content available remote Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows
80%
Downarowicz T. , Lacroix
Studia Mathematica
|
1998
|
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.
11
Content available remote Metoda Aveza i jej uogólnienia
80%
Dawidowicz A. L.
Matematyka Stosowana : matematyka dla społeczeństwa
|
2007
|
tom Nr 8
46-55
PL
W niniejszej pracy przedstawione są twierdzenia dotyczące istnienia miar niezmienniczych ze szczególnym uwzględnieniem metody Aveza, której zastosowania były przedmiotem prac prof. Lasoty.
EN
In this paper the method of construction of invariant measure are presented. Particularly the method of Avez is presented. This method was used by Professor Lasota.
12
Content available 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.
13
Content available remote Ergodic properties of skew products withfibre maps of Lasota-Yorke type
80%
Kowalski Z. S.
Applicationes Mathematicae
|
1993-1995
|
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.
14
Content available remote 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.
15
Content available remote The ergodic decomposition defined by actions of amenable groups, part II: more about the role played by the ergodic invariant probability measures in the decomposition
70%
Zaharopol R.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2021
|
tom Vol. 69, no. 1
61--68
EN
Our purpose here is to continue the study of the ergodic decomposition for actions defined by amenable groups, started in [R. Zaharopol, Colloq. Math. 165 (2021)]. We consider the set Γ(w)αcpie defined in the above-mentioned paper, and we prove that it is Borel measurable and of maximal probability.
16
Content available remote Upper estimate of concentration and thin dimensions of measures
70%
Gacki H. , Lasota A. , Myjak J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2009
|
tom Vol. 57, no 2
149-162
EN
We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.
17
Content available remote Continuous Iterated Function Systems on Polish spaces
70%
Horbacz K. , Szarek T.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2001
|
tom Vol. 49, no 2
191-202
EN
Continuous Iterated Function Systems are studied. We generalize results proved by A. Lasota and RM. C. Mackey to the case when the systems are defined on Polish spaces.
18
Content available remote On nearly selfoptimizing strategies for multiarmed bandit problems with controlled arms
70%
Drabik E.
Applicationes Mathematicae
|
1995-1996
|
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.
19
Content available remote Exploring the Infinite-Time Behavior of the Chaos Game : Approximation and Interactive Visualization of 3D IFSP and RIFS Invariant Measures Using PC Graphics Accelerators
70%
Martyn T.
Machine Graphics and Vision
|
2009
|
tom Vol. 18, No. 4
453-476
EN
In this paper we tackle the problem of approximation and visualization of invariant measures arising from Iterated Function Systems with Probabilities (IFSP) and Recurrent Iterated Function Systems (RIFS) on R³. The measures are generated during the evolution of a stochastic dynamical system, which is a random process commonly known as the chaos game. From the dynamical system viewpoint, an invariant measure gives a temporal information on the long-term behavior of the chaos game related to a given IFSP or RIFS. The non-negative number that the measure takes on for a given subset of space says how often the dynamical system visits that subset during the temporal evolution of the system as time tends to infinity. In order to approximate the measures, we propose a method of measure instancing that can be considered an analogue of object instancing for IFS attractors. Although the IFSP and RIFS invariant measures are generated by the long-term behavior of stochastic dynamical systems, measure instancing makes it possible to compute the value that the measure takes on for a given subset of space in a deterministic way at any accuracy required. To visualize the data obtained with the algorithm, we use direct volume rendering. To incorporate the global structure of invariant measures along with their local properties in an image, a modification of a shading model based on varying density emitters is used. We adapt the model to match the fractal measure context. Then we show how to implement the model on commodity graphics hardware using an approach that combines GPU-based direct volume raycasting and 3D texture slicing used in the object-aligned manner. By means of the presented techniques, visual exploration of 3D IFSP and RIFS measures can be carried out efficiently at interactive frame rates.
20
On measurable Sierpiński-Zygmund functions
60%
Kharazishvili A. B.
Journal of Applied Analysis
|
2006
|
tom Vol. 12, nr 2
283-292
EN
It is proved that there exists a Sierpiński-Zygmund function, which is measurable with respect to a certain invariant extension of the Lebesgue measure on the real line R.
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