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Języki publikacji
Abstrakty
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.
Czasopismo
Rocznik
Tom
Strony
471--480
Opis fizyczny
Bibliogr. 8 poz.
Twórcy
autor
- via Dazzi, 11 50141 Firenze, Italy, gerald.goodman@gmail. com
Bibliografia
- [1] S. Graf, On Bandt’s tangential distribution for self-similar measures, Monatsh. Math. 120 (1995) 3–4, 223–246.
- [2] G.S. Goodman, Fractal sets satisfying the strong open set condition in complete metric spaces, Opusc. Math. 28 (2008) 4, 463–470.
- [3] J.E. Hutchinson, Fractals and self-similarity, Indiana J. Math. 30 (1981), 713–747.
- [4] S.P. Lalley, The packing and covering dimensions of some self-similar fractals, Indiana Univ. Math. J. 37 (1988), 699–709.
- [5] A. Lasota, J. Myjak, and T. Szarek, Markov operators with a unique invariant measure, J. Math. Anal. Appl. 276 (2002), 343–356.
- [6] M. Moran and H.-M, Rey, Singularity of self-similar measures with respect to Hausdorff measures, Trans. A.M.S. 350 (1998), 2297–2310.
- [7] N. Patzschke, Self-conformal multifractal measures, Adv. Appl. Math. 10 (1997), 487–513.
- [8] W. Sierpinski, General Topology, University of Toronto Press, Toronto, 1952.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGH9-0005-0009