# Markov operators on the space of vector measures; coloured fractals

Annales Polonici Mathematici (1998)

- Volume: 69, Issue: 3, page 217-234
- ISSN: 0066-2216

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topKarol Baron, and Andrzej Lasota. "Markov operators on the space of vector measures; coloured fractals." Annales Polonici Mathematici 69.3 (1998): 217-234. <http://eudml.org/doc/270485>.

@article{KarolBaron1998,

abstract = {We consider the family 𝓜 of measures with values in a reflexive Banach space. In 𝓜 we introduce the notion of a Markov operator and using an extension of the Fortet-Mourier norm we show some criteria of the asymptotic stability. Asymptotically stable Markov operators can be used to construct coloured fractals.},

author = {Karol Baron, Andrzej Lasota},

journal = {Annales Polonici Mathematici},

keywords = {vector measures; Fortet-Mourier norm; Markov operators; asymptotic stability; iterated function systems; IFS; coloured fractals},

language = {eng},

number = {3},

pages = {217-234},

title = {Markov operators on the space of vector measures; coloured fractals},

url = {http://eudml.org/doc/270485},

volume = {69},

year = {1998},

}

TY - JOUR

AU - Karol Baron

AU - Andrzej Lasota

TI - Markov operators on the space of vector measures; coloured fractals

JO - Annales Polonici Mathematici

PY - 1998

VL - 69

IS - 3

SP - 217

EP - 234

AB - We consider the family 𝓜 of measures with values in a reflexive Banach space. In 𝓜 we introduce the notion of a Markov operator and using an extension of the Fortet-Mourier norm we show some criteria of the asymptotic stability. Asymptotically stable Markov operators can be used to construct coloured fractals.

LA - eng

KW - vector measures; Fortet-Mourier norm; Markov operators; asymptotic stability; iterated function systems; IFS; coloured fractals

UR - http://eudml.org/doc/270485

ER -

## References

top- [1] M. F. Barnsley, Fractals Everywhere, Academic Press, 1988. Zbl0691.58001
- [2] M. F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A 399 (1985), 243-275. Zbl0588.28002
- [3] J. Diestel and J. J. Uhl, Jr., Vector Measures, Amer. Math. Soc., 1977.
- [4] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. Zbl0598.28011
- [5] A. Lasota, From fractals to stochastic differential equations, in: P. Garbaczewski, M. Wolf and A. Weron (eds.), Chaos - The Interplay Between Stochastic and Deterministic Behaviour, Springer, 1995, 235-255. Zbl0835.60058
- [6] A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Springer, 1994.
- [7] A. Lasota and J. A. Yorke, Lower bound technique for Markov operators and iterated function systems, Random Comput. Dynam. 2 (1994), 41-77. Zbl0804.47033
- [8] A. A. Markov, Extension of the law of large numbers for dependent variables, Izv. Fiz.-Mat. Obshch. Kazansk. Univ. (2) 15 (1906), 135-156 (in Russian).
- [9] E. Nummelin, General Irreducible Markov Chains and Non-negative Operators, Cambridge Univ. Press, 1984. Zbl0551.60066
- [10] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, 1967. Zbl0153.19101
- [11] S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, Wiley, 1991. Zbl0744.60004

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