TY  - JOUR
ID  - Goodman2008
LB  - Goodman2008
AU  - Goodman, Gerald S.
TI  - Fractal sets satisfying the strong open set condition in complete metric spaces
ST  - Fractal sets satisfying the strong open set condition in complete metric spaces
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2008
VL  - 28
IS  - 4
SP  - 463
EP  - 470
DO  - http://dx.doi.org/10.7494/OpMath.2008.28.4.463
UR  - http://dx.doi.org/10.7494/OpMath.2008.28.4.463
AB  - 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.
KW  - address
KW  - Baire category
KW  - fractal
KW  - scaling function
KW  - scaling operator
KW  - strong open set condition
SN  - 1232-9274
ER  -