On dimensions of visible parts of self-similar sets with finite rotation groups
- 24 March 2022
- journal article
- research article
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 150 (7), 2983-2995
- https://doi.org/10.1090/proc/15843
Abstract
We derive an upper bound for the Assouad dimension of visible parts of self-similar sets generated by iterated function systems with finite rotation groups and satisfying the weak separation condition. The bound is valid for all visible parts and it depends on the direction and the penetrable part of the set, which is a concept defined in this paper. As a corollary, we obtain in the planar case that if the projection is a finite or countable union of intervals then the visible part is 1-dimensional. We also prove that the Assouad dimension of a visible part is strictly smaller than the Hausdorff dimension of the set provided the projection contains interior points. Our proof relies on Furstenberg’s dimension conservation principle for self-similar sets.Keywords
Funding Information
- Academy of Finland (Centre of Excellence in Analysis and Dynamics Research, Centre of Excellence in Analysis and Dynamics Research, 318217)
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