Invariance of Borel Classes in Metric Spaces.
Let (with ) be vector fields of class in an open set , let be a -dimensional submanifold of and define where is the tangent space to at . Then we expect the following property, which is obvious in the special case when is an interior point (relative to ) of : If is a -density point (relative to ) of then all the iterated Lie brackets of order less or equal to
Contractive Markov systems on Polish spaces which arise from graph directed constructions of iterated function systems with place dependent probabilities are considered. It is shown that their stability may be studied using the concentrating methods developed by the second author [Dissert. Math. 415 (2003)]. In this way Werner's results obtained in a locally compact case [J. London Math. Soc. 71 (2005)] are extended to a noncompact setting.
Nonoverlapping contractive self-similar iterated function systems (IFS) have been studied in great detail via the open set condition. On the other hand much less is known about IFS with overlaps. To deal with such systems, a weak separation condition (WSC) has been introduced recently [LN1]; it is weaker than the open set condition and it includes many important overlapping cases. This paper has two purposes. First, we consider the class of self-similar measures generated by such IFS; we give a...