Measures with predetermined regularity and inhomogeneous self-similar sets
Käenmäki, A., & Lehrbäck, J. (2017). Measures with predetermined regularity and inhomogeneous self-similar sets. Arkiv för Matematik, 55(1), 165-184. https://doi.org/10.4310/ARKIV.2017.v55.n1.a8
Published inArkiv för Matematik
© 2017 by Institut Mittag-Leffler.
We show that if X is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of X as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous self-similar set EC coincides with the lower dimension of the condensation set C, while the Assouad dimension of EC is the maximum of the Assouad dimensions of the corresponding self-similar set E and the condensation set C. If the Assouad dimension of C is strictly smaller than the Assouad dimension of E, then the upper regularity dimension of any measure supported on EC is strictly larger than the Assouad dimension of EC. Surprisingly, the corresponding statement for the lower regularity dimension fails.
Publication in research information system
MetadataShow full item record
Showing items with similar title or keywords.
Le Donne, Enrico; Rajala, Tapio (Indiana University, 2015)We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper ...
Lehrbäck, Juha (Hebrew University Magnes Press; Springer, 2017)We establish both sufficient and necessary conditions for weighted Hardy inequalities in metric spaces in terms of Assouad (co)dimensions. Our sufficient conditions in the case where the complement is thin are new, even ...
Rajala, Kai; Rasimus, Martti; Romney, Matthew (Springer, 2021)We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to R2R2. Given a measure μμ on such a space, we introduce μμ-quasiconformal maps f:X→R2f:X→R2, ...
Heino, Joonas (International Statistical Institute; Bernoulli Society for Mathematical Statistics and Probability, 2018)We show that a uniform measure density condition implies game regularity for all 2 < p < ∞ in a stochastic game called “tug-of-war with noise”. The proof utilizes suitable choices of strategies combined with estimates for ...
Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy-Sobolev -type Inequalities Dyda, Bartłomiej; Ihnatsyeva, Lizaveta; Lehrbäck, Juha; Tuominen, Heli; Vähäkangas, Antti (Springer, 2019)