Finite energy coordinates and vector analysis on fractals

TL;DR

This paper develops a framework for defining and analyzing vector calculus concepts like gradient, divergence, and tangent spaces on fractals and other irregular spaces using energy finite coordinates associated with Dirichlet forms.

Contribution

It introduces coordinate formulas for tangent spaces, gradient, divergence, and generators on fractals and irregular spaces, extending classical differential calculus to these settings.

Findings

Formulated coordinate systems for fractals and irregular spaces.

Derived gradient, divergence, and generator formulas in these coordinates.

Applied framework to examples like Euclidean spaces, Heisenberg group, and Sierpinski gasket.

Abstract

We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian local charts, domains on the Heisenberg group and the measurable Riemannian geometry on the Sierpinski gasket.