SEMINARIO DI MATEMATICA APPLICATA (11 marzo)

[Giacomo Aletti <giacomo.aletti@unimi.it>]

Martedi' 11 Marzo 2003, ore 16.00
Aula Rappresentanza, Dipartimento di Matematica (Via Saldini 50, Milano)



SEMINARIO DI MATEMATICA APPLICATA
---------------------------------


Fractal Measures whose Supports satisfy the
SOSC


Gerald S. Goodman

University of Florence
Statistics Department
Viale Morgagni, 59
I-50134 Florence


The Hutchinson fractal $K$ associated with the scaling maps $w_i$,
$i=1,\ldots, N$, satisfies the Strong Open Set Condition, SOSC, if
it intersects a non-empty open set $V$ having the properties
\[
w_i \left(V\right) \subset V, \, \mbox{ for each } i,
\]
\[
w_i\left(V\right) \cap w_j \left(V\right)= \emptyset, \mbox {
whenever } i \neq j.
\]
Define the \emph{core} $\check{K}$ of $K$, as
\[
\check{K}=\bigcap^\infty_{n=0} S^n \left(V\right),
\]
where $S^n$ is the $n$-th iterate of the scaling operator
$S=\bigcup^{N}_{i=1} w_i$. We show that when the SOSC holds and
the restrictions of the scaling maps to $V$ are open, the core is
a non-empty, dense subset of $K$, invariant under $S$. The same is
also true of $K \backslash \check{K}$, provided the scaling maps
are injective. A direct construction establishes that, if the SOSC
holds, measures on $K$ that are invariant under the Markov
operator are necessarily multiplicative, and they assign full
measure to $\check{K}$. Consequently, they vanish on the
intersection of sets of the form $w_{i_i} \circ \cdots \circ
w_{i_n} \left(K\right)$, generalizing a known result for scaling
maps that are strictly self-similar.




---------------------------------
La home page del seminario e' http://www.users.unimi.it/~pavarino/sma/
Il seminario e' organizzato dal Dipartimento di Matematica dell'Universita'
di Milano in collaborazione con MIRIAM (MIlan Research group in Industrial
and Applied Mathematics) e il Dottorato di Ricerca MACRO (MAtematica
Computazionale e Ricerca Operativa) e MSSCI (Matematica, Statistica, Scienze
Computazionali e Informatica). Per ulteriori informazioni contattare Luca
Pavarino o Kevin Payne.




--------------------------

Giacomo Aletti, Ph.D.
Dept. of Mathematics
University of Milan
V. Saldini, 50
20131 Milano (MI)
ITALY
Tel. +39(02)-503-16158
Fax +39(02)-503-16092
giacomo.aletti@unimi.it