6533b85bfe1ef96bd12ba111

RESEARCH PRODUCT

Measures with predetermined regularity and inhomogeneous self-similar sets

Antti KäenmäkiJuha Lehrbäck

subject

Pure mathematicsAssouad dimensionGeneral MathematicsOpen set01 natural sciencesMeasure (mathematics)Complete metric space54E35010305 fluids & plasmasSet (abstract data type)Dimension (vector space)0103 physical sciencesClassical Analysis and ODEs (math.CA)FOS: Mathematicsinhomogeneous self-similar setMathematics::Metric Geometry28A200101 mathematicsMathematics010102 general mathematicsta111doubling metric space54F45lower dimensionMathematics - Classical Analysis and ODEs28A75uniform perfectness

description

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.

yearjournalcountryeditionlanguage
2016-09-12
https://dx.doi.org/10.48550/arxiv.1609.03325