Abstract
We study the analogue of a convex interpolant of two sets on Sierpinski gaskets and an associated notion of measure transport. The structure of a natural family of interpolating measures is described and an interpolation inequality is established. A key tool is a good description of geodesics on these gaskets, some results on which have previously appeared in the literature.
Highlights
The study of functional inequalities in the setting of fractal metric-measure spaces is considerably less developed
In the case of certain fractal sets and sets with fractal boundary in Euclidean space, one achievement of the theory developed by Lapidus and collaborators is a characterization of the volume of -neighborhoods using complex dimensions, which in turn are connected to analytic structure on the set through the zeta function of its Laplacian [14]
The purpose of this paper is to consider the elementary notion of convex interpolant in the setting of one well-studied class of fractals, the Sierpinski gaskets Sn defined on regular n-simplices in Rn
Summary
The Sierpinski n-gasket Sn ⊆ Rn is the unique nonempty compact attractor of the iterated function system (IFS). A measure μn of this type is a probability measure determined uniquely from a set of weights {μin}ni=0, where each μin > 0 and i μin = 1, by the requirement that for any measurable X ⊆ Sn one has the self-similar identity μn(X) = μinμn(Fi−1(X)). The measure on interpolant sets will be a projection of the original self-similar measure, and will be self-similar itself This is a consequence of the following results, which are well-known when each Fi is a homothety, as it is here. In the setting of Lemma 1.2, let K denote the attractor of the IFS and μ be the self-similar measure on K with weights μi. The pushforward measure φ∗μ satisfies the self-similar identity φ∗μ(X) = μiφ∗μ(Fi−1(X)).
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