Browsing by Subject "scaling function"

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Access status: Restricted ,
Falki o nośniku zwartym
(Data obrony: 2014-10-30) Nikiel, Paweł
Wydział Matematyki Stosowanej
Access status: Open Access ,
Fractal sets satisfying the strong open set condition in complete metric spaces
(2008) Goodman, Gerald S.
Let $K$ be a Hutchinson fractal in a complete metric space $X$, invariant under the action $S$ of the union of a finite number of Lipschitz contractions. The Open Set Condition states that $X$ has a non-empty subinvariant bounded open subset $V$, whose images under the maps are disjoint. It is said to be strong if $V$ meets $K$. We show by a category argument that when $K \not\subset V$ and the restrictions of the contractions to $V$ are open, the strong condition implies that $\check{V}=\bigcap_{n=0}^{\infty} S^n(V)$, termed the core of $V$, is non-empty. In this case, it is an invariant, proper, dense, subset of $K$, made up of points whose addresses are unique. Conversely, $\check{V}\neq \emptyset$ implies the SOSC, without any openness assumption.
Access status: Open Access ,
Invariant measures whose supports possess the strong open set property
(2008) Goodman, Gerald S.
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 $\mu _P$ 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 $V \subset X$ exists whose images under the maps are disjoint; it is strong if $K \cap V \neq \emptyset$. In that case, the core of $V$, $\check{V}=\bigcap_{n=0}^{\infty} S^n (V)$, is non-empty and dense in $K$. Moreover, when $X$ is separable, $\check{V}$ has full $\mu _P$-measure for every $P$. We show that the strong condition holds for $V$ satisfying the OSC iff $\mu_P (\partial V) =0$, and we prove a zero-one law for it. We characterize the complement of $\check{V}$ 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.
Access status: Restricted ,
Zastosowanie procesów Levy'ego do wyceny kontraktów forward na towary
(Data obrony: 2015-10-16) Budzińska, Małgorzata
Wydział Matematyki Stosowanej