Tangential dimensions I. Metric spaces

TL;DR

This paper introduces pointwise tangential dimensions for metric spaces, providing a tool to analyze local multifractal behavior and oscillations in dimensions at specific points, with applications in noncommutative geometry.

Contribution

It defines tangential dimensions based on tangent sets and demonstrates their sensitivity to local oscillations in fractal dimensions, extending concepts from noncommutative geometry.

Findings

Upper and lower tangential dimensions can differ significantly.

Tangential dimensions detect oscillations not visible through classical box dimensions.

Examples show the divergence of tangential dimensions even when local box dimensions agree.

Abstract

Pointwise tangential dimensions are introduced for metric spaces. Under regularity conditions, the upper, resp. lower, tangential dimensions of X at x can be defined as the supremum, resp. infimum, of box dimensions of the tangent sets, a la Gromov, of X at x. Our main purpose is that of introducing a tool which is very sensitive to the "multifractal behaviour at a point" of a set, namely which is able to detect the "oscillations" of the dimension at a given point. In particular we exhibit examples where upper and lower tangential dimensions differ, even when the local upper and lower box dimensions coincide. Tangential dimensions can be considered as the classical analogue of the tangential dimensions for spectral triples introduced in math.OA/0202108 and math.OA/0404295, in the framework of Alain Connes' noncommutative geometry.