Iterated function systems with overlaps and self-similar measures. (English) Zbl 1019.28005
The authors consider the system of iterates of a finite number of similitudes with the same contraction ratio in the Euclidean space. Overlappings in the images of the similitudes are allowed, but still verifying a weak separation assumption, generalizing the results for systems under the strong open set condition.
The paper focusses on the analysis of the self-similar measure supported in the invariant subset associated with the iterated function system, which is either absolutely continuous or singular. The theorems state several conditions to determine that dichotomy: mainly a sufficient assumption for the singularity of the measure, and an equivalent condition for its absolute continuity.
The paper focusses on the analysis of the self-similar measure supported in the invariant subset associated with the iterated function system, which is either absolutely continuous or singular. The theorems state several conditions to determine that dichotomy: mainly a sufficient assumption for the singularity of the measure, and an equivalent condition for its absolute continuity.
Reviewer: Eleonora Catsigeras (Montevideo)
MSC:
28A80 | Fractals |
37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
37A05 | Dynamical aspects of measure-preserving transformations |
28D05 | Measure-preserving transformations |