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1
Content available remote Twierdzenia Ponceleta w interpretacji Rafała Kołodzieja
Żołądek H.
Wiadomości Matematyczne
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2013
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T. 49, Nr 2
29--45
PL
Głównym celem tego artykułu jest zaprezentowanie ciekawej konstrukcji miary niezmienniczej dla pewnego przekształcenia elipsy w siebie. Konstrukcja ta pochodzi od Rafała Kołodzieja (1956-2011) i znalazła zastosowanie w pewnych klasycznych geometrycznych zagadnieniach zapoczątkowanych badaniami dziewiętnastowiecznego matematyka francuskiego Jeana-Victora Ponceleta. [...]
2
Content available remote Lower bound technique and its applications to function systems and stochastic partial differential equations
Szarek T.
Mathematica Applicanda
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2013
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Vol. 41, No. 2
185--198
EN
We formulate some criteria for the existence of an invariant measure for Markov chains and Markov processes. We also show their application in the theory of function systems and stochastic differential equations
PL
W pracy formułujemy kryteria dla istnienia miary niezmienniczej dla łańcuchów i procesów Markowa. Następnie pokazujemy ich użyteczność w teorii iterowanych układów funkcyjnych i stochastycznych równań rózniczkowych.
3
Content available Ergodic theory approach to chaos: remarks and computational aspects
Mitkowski P. J., Mitkowski W.
International Journal of Applied Mathematics and Computer Science
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2012
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Vol. 22, no. 2
259-267
EN
We discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the previously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor of a nonsimple structure, supporting an invariant mixing measure. This observation verifies Lasota's conjecture concerning nontrivial ergodic properties of the model.
4
Content available remote Topological pressure for one-dimensional holomorphic dynamical systems
Gelfert K., Wolf Ch.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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Vol. 55, no 1
53-62
EN
For a class of one-dimensional holomorphic maps ƒ of the Riemann sphere we prove that for a wide class of potentials φ the topological pressure is entirely determined by the values of φ on the repelling periodic points of ƒ. This is a version of a classical result of Bowen for hyperbolic diffeomorphisms in the holomorphic non-uniformly hyperbolic setting.
5
Content available remote On a dimension of measures
Lasota A., Myjak J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2002
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Vol. 50, no 2
221-235
EN
Using the notion of the Levy concentration function we discuss a definition of the dimension for probability measures. This dimension is strongly connected with the correlation dimension of measures and with the Hausdorff dimension of sets. Moreover, we calculate some bounds of this dimension for measures generated by Iterated Function Systems and by a partial differential equation.
6
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.
7
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.
8
Content available remote Attractors of multifunetions
Lasota A., Myjak J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2000
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Vol. 48, no 3
319-334
EN
We introduce the notion of a semiattractor and attractor for multifunctions and we show that they have properties similar to semifractals and fractals. Further we show a relationship between the multifunctions and transition functions appearing in the theory of Markov operators.
9
Content available remote Semifractals on Polish spaces
Lasota A., Myjak J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
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Vol. 46, no 2
179-196
EN
We extend the theory of semifractals to arbitrary metric spaces. We also show kow to construct semifractals on Polish spaces by a use of Markov operators and Markov chain.
10
Content available remote Limit points of random iterations of monotone maps
Sikora R.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
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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.
11
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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