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.
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