University of Oulu

Järvenpää, E., Järvenpää, M., Suomala, V., & Wu, M. (2022). On dimensions of visible parts of self-similar sets with finite rotation groups. Proceedings of the American Mathematical Society, 150(7), 2983–2995.

On dimensions of visible parts of self-similar sets with finite rotation groups

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Author: Järvenpää, Esa1; Järvenpää, Maarit1; Suomala, Ville1;
Organizations: 1Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.2 MB)
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Language: English
Published: American Mathematical Society, 2022
Publish Date: 2022-05-20


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.

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Series: Proceedings of the American Mathematical Society
ISSN: 0002-9939
ISSN-E: 1088-6826
ISSN-L: 0002-9939
Volume: 150
Pages: 2983 - 2995
DOI: 10.1090/proc/15843
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: The fourth author was supported by the Academy of Finland, project grant No. 318217. This study was supported by the Centre of Excellence in Analysis and Dynamics Research funded by the Academy of Finland.
Academy of Finland Grant Number: 318217
Detailed Information: 318217 (Academy of Finland Funding decision)
Copyright information: © Copyright 2022 American Mathematical Society.