TY  - JOUR
ID  - Goodman2008
LB  - Goodman2008
AU  - Goodman, Gerald S.
TI  - Invariant measures whose supports possess the strong open set property
ST  - Invariant measures whose supports possess the strong open set property
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2008
VL  - 28
IS  - 4
SP  - 471
EP  - 480
DO  - http://dx.doi.org/10.7494/OpMath.2008.28.4.471
UR  - http://dx.doi.org/10.7494/OpMath.2008.28.4.471
AB  - 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.
KW  - core
KW  - fractal
KW  - fractal measure
KW  - invariant measure
KW  - scaling function
KW  - scaling operator
KW  - strong open set condition
KW  - zero-one law
SN  - 1232-9274
ER  -