Tangential dimensions II. Measures

TL;DR

This paper introduces a new concept of tangential dimensions for measures in R^n, capable of detecting local oscillations and multifractal behavior, with explicit calculations on translation fractals and connections to noncommutative geometry.

Contribution

It defines tangential dimensions that are sensitive to multifractal oscillations and demonstrates their calculation on translation fractals, linking them to existing geometric and noncommutative frameworks.

Findings

Tangential dimensions can detect oscillations even when local dimensions coincide.

Explicit formulas for translation fractals show constant tangential dimensions.

Connections established between tangential dimensions and noncommutative geometric measures.

Abstract

Notions of (pointwise) tangential dimension are considered, for measures of R^n. Under regularity conditions (volume doubling), the upper resp. lower dimension at a point x of a measure can be defined as the supremum, resp. infimum, of local dimensions of the measures tangent to the given measure at x. Our main purpose is that of introducing a tool which is very sensitive to the "multifractal behaviour at a point" of a measure, namely which is able to detect the "oscillations" of the dimension at a given point, even when the local dimension exists, namely local upper and lower dimensions coincide. These definitions are tested on a class of fractals, which we call translation fractals, where they can be explicitly calculated for the canonical limit measure. In these cases the tangential dimensions of the limit measure coincide with the metric tangential dimensions of the fractal defined in math.FA/0305091, and they are constant, i.e. do not depend on the point. However, upper and lower dimensions may differ. Moreover, on these fractals, these quantities coincide with their noncommutative analogues, defined in math.OA/0202108 and math.OA/0404295, in the framework of Alain Connes' noncommutative geometry.