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1
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
Zaharopol Radu
Bulletin of the Polish Academy of Sciences. Mathematics
|
2021
|
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.
2
Content available remote Random Iteration with Place Dependent Probabilities
Kapica Rafał, Ślęczka Maciej
Probability and Mathematical Statistics
|
2020
|
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.
3
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
Martyn T.
Machine Graphics and Vision
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2009
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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.
4
Content available remote Upper estimate of concentration and thin dimensions of measures
Gacki H., Lasota A., Myjak J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2009
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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.
5
Content available The work of professor Andrzej Lasota on asymptotic stability and recent progress
Bartoszek W.
Opuscula Mathematica
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2008
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Vol. 28, no. 4
395-413
EN
The paper is devoted to Professor Andrzej Lasota's contribution to the ergodic theory of stochastic operators. We have selected some of his important papers and shown their influence on the evolution of this topic. We emphasize the role A. Lasota played in promoting abstract mathematical theories by showing their applications. The article is focused exclusively on ergodic properties of discrete stochastic semigroups {Pn : n ≥ 0}. Nevertheless, almost all of Lasota's results presented here have their one-parameter continuous semigroup analogs.
6
Content available Invariant measures whose supports possess the strong open set property
Goodman G. S.
Opuscula Mathematica
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2008
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Vol. 28, no. 4
471-480
EN
Let X be a complete metric space, and S the union of a finite number of strict contractions on it. If P is a probability distribution on the maps, and K is the fractal determined by S, there is a unique Borel probability measure µp on X which is invariant under the associated Markov operator, and its support is K. The Open Set Condition (OSC) requires that a non-empty, subinvariant, bounded open set V⊂ X exists whose images under the maps are disjoint; it is strong if K ∩ V ≠ 0.In that case, the core of [formula] is non-empty and dense in K. Moreover, when X is separable, V has full µp-measure for every P. We show that the strong condition holds for V satisfying the OSC iff µp(ϑV) = 0, and we prove a zero-one law for it. We characterize the complement of V relative to K, and we establish that the values taken by invariant measures on cylinder sets defined by K, or by the closure of V, form multiplicative cascades.
7
Content available remote Metoda Aveza i jej uogólnienia
Dawidowicz A.L.
Matematyka Stosowana : matematyka dla społeczeństwa
|
2007
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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.
8
On measurable Sierpiński-Zygmund functions
Kharazishvili A. B.
Journal of Applied Analysis
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2006
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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.
9
Some results of ergodic theory and their application to drilling modeling
Chakvetadze G.
Prace Naukowe Instytutu Matematyki Politechniki Wrocławskiej. Konferencje
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2001
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Vol. 24, nr 3
7-16
10
Content available remote Continuous Iterated Function Systems on Polish spaces
Horbacz K., Szarek T.
Bulletin of the Polish Academy of Sciences. Mathematics
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2001
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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.
11
Content available remote The dimension of self-similar measures
Szarek T.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2000
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Vol. 48, no 3
293-302
EN
We consider Iterated Function Systems on Polish spaces. The Hausdorff dimension of invariant distributions for such systems is estimated.
12
Content available remote Limit points of random iterations of monotone maps
Sikora R.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
|
Vol. 46, no 4
351-363
EN
We consider random iterations of a monotone, continous map f of a partially ordered compact Polish space. We prove that, under additional assumptions, all limit point of the trajectories are fixed points of f a.s. If the fixed points are isolated, then almost all trajectories are convergent. Invariant measure are supported on the sets of fixed points.
13
Content available remote An application of the Kantorovich-Rubinstein maximum principle in the stability theory of Markov operators
Gacki H.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
|
Vol. 46, no 2
215-223
EN
We consider the asymptotic behaviour of Markov operators acting on measures, defined on a locally are [sigma]-compact metric space. We prove a new sufficient condition for the asymptotic stability of Markov operators. This condition is applied to stochastically peturbed dynamical systems, and iterated function system.
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