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8 Bulletin of the Polish Academy of Sciences. Mathematics

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Infinite iterated function systems : a multivalued approach

Leśniak K.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 1

1-8

We prove that a compact family of bounded condensing multifunctions has bounded condensing set-theoretic union. Compactness is understood in the sense of the Chebyshev uniform semimetric induced by the Hausdorff distance and condensity is taken w.r.t. the Hausdorff measure of noncompactness. As a tool, we present an estimate for the measure of an infinite union. Then we apply our result to infinite iterated function systems.

On a dimension of measures

Lasota A., Myjak J.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 2

221-235

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.

The dimension of self-similar measures

Szarek T.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 3

293-302

We consider Iterated Function Systems on Polish spaces. The Hausdorff dimension of invariant distributions for such systems is estimated.

Attractors of multifunetions

Lasota A., Myjak J.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 3

319-334

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.

Generic properties of continuous iterated function systems

Szarek T.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 1

77-89

It is shown that in the space of all nonexpansive continuous IFS's defined on a compact convex subset of R^n the family of asymptotically stable IFS's having a singular stationary distribution is generic.

A locally connected continuum which is not an IFS attractor

Kwieciński M.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 2

127-132

We construct a locally connected continuum embedded in the plane which is not the attractor of any iterated function system.

Semifractals on Polish spaces

Lasota A., Myjak J.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 2

179-196

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.

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

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.