Tangential dimensions I. Metric spaces

Details Tangential dimensions I. Metric spaces Tangential dimensions I. Metric spaces

2003-05-06

arxiv.org/abs/math/0305091v2

Houston Journal Math., 31 (2005) no. 4, 1023-1045.

Mathematics

Functional Analysis

18 pages, 4 figures. This version corresponds to the first part of v1, the second part being now included in math.FA/0405174

Scientific paper

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.

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