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Towards Analysis on Fractals: Piecewise C^1-Fractal Curves, Spectral Triples, and the Gromov-Hausdorff Propinquity

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Many important physical processes can be described by differential equations. The solutions of such equations are often formulated in terms of operators on smooth manifolds. A natural question is to determine whether differential structures defined on fractals can be realized as a metric limit of differential structures on their approximating finite graphs. One of the fundamental tools of noncommutative geometry is Alain Connes’ spectral triple. Because spectral triples generalize differential structure, they open up promising avenues for extending analytic methods from mathematical physics to fractal spaces. The Gromov-Hausdorff distance is an important tool of Riemannian geometry and building on the earlier work of Marc Rieffel, Frederic Latremoliere introduced a generalization of the Gromov-Hausdorff distance that was recently extended to spectral triples. The class of piecewise $C^1$-fractal curves was first characterized by Michel Lapidus and Jonathan Sarhad as a generalized setting for the spectral triple construction developed by Christensen, Ivan, and Lapidus in the context of the Sierpinski gasket. We provide an analytic framework for the metric approximation of the Lapidus-Sarhad spectral triple on a piecewise $C^1$-fractal curve by spectral triples defined on an approximating sequence of finite graphs which exhibit properties motivated by the setting of the Sierpinski gasket.

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