Measures with predetermined regularity and inhomogeneous self-similar sets

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 $E_C$ coincides with the lower dimension of the condensation set $C$, while the Assouad dimension of $E_C$ 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 $E_C$ is strictly larger than the Assouad dimension of $E_C$. Surprisingly, the corresponding statement for the lower regularity dimension fails.

Keywords

doubling metric space, uniform perfectness, Assouad dimension, lower dimension, inhomogeneous self-similar set

2010 Mathematics Subject Classification

Primary 28A75, 54E35. Secondary 28A20, 54F45.

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JL has been supported in part by the Academy of Finland (project #252108).

Received 21 September 2016

Accepted 10 April 2017

Published 26 September 2017