[Paper Review] Tangential dimensions II. Measures

Daniele Guido, Tommaso Isola|ArXiv.org|May 10, 2004

Mathematical Dynamics and Fractals8 references5 citations

TL;DR

This paper introduces tangential dimensions for measures on $ℝ^N$ as a tool to detect local oscillations in fractal behavior, defining upper and lower tangential dimensions as the suprema and infima of local dimensions of tangent measures under volume doubling. The key contribution is that these dimensions precisely capture multifractal oscillations and coincide with metric and noncommutative tangential dimensions on translation fractals, where they are constant across points despite potential differences between upper and lower dimensions.

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.

Motivation & Objective

• To develop a dimension theory sensitive to local oscillations in the fractal behavior of measures, especially when local dimension fails to converge.
• To define upper and lower tangential dimensions of a measure at a point as suprema and infima of local dimensions of tangent measures.
• To establish conditions under which tangential dimensions of measures coincide with those of the underlying metric space and with noncommutative geometric counterparts.
• To compute these dimensions explicitly for canonical limit measures on translation fractals, where they are constant and match known geometric and spectral invariants.

Proposed method

• Define upper and lower tangential dimensions via iterated logarithmic ratios of measure masses over nested balls: $\underline{\delta}_{\mu}(x) = \liminf_{\lambda\to 0} \liminf_{r\to 0} \frac{\log(\mu(B(x,r))/\mu(B(x,\lambda r)))}{\log 1/\lambda}$.
• Introduce a function $f(t) = -\log \mu(B(x,e^{-t}))$ and re-express tangential dimensions in terms of differences $g(t,h) = f(t+h) - f(t)$, enabling analysis via asymptotic growth rates.
• Establish equivalence between tangential dimensions and local dimensions of tangent measures under the volume doubling condition.
• Use the framework of translation fractals—self-similar sets with open set condition—where the canonical limit measure satisfies the required regularity, enabling explicit computation.
• Relate the tangential dimensions to the spectral triple invariants from noncommutative geometry, showing full agreement with results from [5,6].
• Apply asymptotic analysis to sequences $P_n$ and $\Lambda_n$ representing measure mass and scale at level $n$, deriving expressions for upper and lower local dimensions as liminf/limsup of $\log P_n / \log 1/\Lambda_n$.

Experimental results

Research questions

• RQ1 Can tangential dimensions detect oscillatory behavior in the local dimension of a measure, even when the local dimension exists?
• RQ2 Under what conditions do the tangential dimensions of a measure coincide with those of the underlying metric space?
• RQ3 Do the tangential dimensions of the canonical limit measure on translation fractals match their noncommutative geometric counterparts?
• RQ4 What is the precise relationship between the tangential dimensions of a measure and the local dimensions of its tangent measures?
• RQ5 How do the tangential dimensions behave on translation fractals, and are they constant across points?

Key findings

• On translation fractals with open set condition, the tangential dimensions of the canonical limit measure are constant across all points and equal to the metric tangential dimensions of the fractal.
• The upper and lower tangential dimensions of the measure may differ, even though they are constant, reflecting oscillatory behavior in the local dimension.
• For translation fractals, the tangential dimensions coincide with the noncommutative tangential dimensions computed via spectral triples in [5,6], establishing a bridge between geometric and noncommutative analysis.
• Under the volume doubling condition, the upper and lower tangential dimensions of a measure are equivalent to the suprema and infima of the local dimensions of its tangent measures.
• The local dimensions $\underline{d}_\mu(x)$ and $\overline{d}_\mu(x)$ are given by $\liminf_{n\to\infty} \frac{\log P_n}{\log 1/\Lambda_n}$ and $\limsup_{n\to\infty} \frac{\log P_n}{\log 1/\Lambda_n}$, respectively, where $P_n$ is the measure mass at scale $\Lambda_n$.
• The Hausdorff dimension $d_H(F)$ equals $\underline{d}_\mu(x)$ if and only if $\liminf(\log P_n - d_H(F) \log 1/\Lambda_n)$ is finite, linking tangential dimensions to the existence of nontrivial Hausdorff measure.