Invariant measures whose supports possess the strong open set property

• Autorzy: Goodman G. S.
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

core; fractal; fractal measure; invariant measure; scaling function; scaling operator; strong open set condition; zero-one law

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2008
• Tom: Vol. 28, no. 4
• Strony: 471--480
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Goodman G. S.

• via Dazzi, 11 50141 Firenze, Italy,

Bibliografia

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