Self-Affine Vector Measures and Vector Calculus on Fractals
In this paper, we construct an IFS framework for studying self-similar vector measures. These measures have several applications, including the tangent and normal vector measure “fields” to fractal curves. Using the tangent vector measure, we define a line integral of a smooth vector field over a fractal curve. This then leads to a formulation of Green’s Theorem (and the Divergence Theorem) for planar regions bounded by fractal curves. The general IFS setting also leads to “probability measure” — valued measures, which give one way to coloring the geometric attractor of an IFS in a self-similar way.
KeywordsVector Measure Iterate Function System Borel Probability Measure Fractal Curve Markov Operator
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