Abstract
It is shown that many important features of nested fractals, such as the Hausdorff dimension and measure, the geodesic distance induced by the immersion in \(\mathbb{R}^n\) (when it exists), and the self-similar energy can be recovered by the description of the fractal in terms of spectral triples. We describe in particular the case of the Vicsek square, showing that all self-similar energies can be described through a deformation of the square to a rhombus.
Mathematics Subject Classification (2010). Primary 58B34; Secondary 28A80, 58J42
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© 2016 Springer International Publishing Switzerland
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Guido, D., Isola, T. (2016). New Results on Old Spectral Triples for Fractals. In: Alpay, D., Cipriani, F., Colombo, F., Guido, D., Sabadini, I., Sauvageot, JL. (eds) Noncommutative Analysis, Operator Theory and Applications. Operator Theory: Advances and Applications(), vol 252. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29116-1_12
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DOI: https://doi.org/10.1007/978-3-319-29116-1_12
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-29114-7
Online ISBN: 978-3-319-29116-1
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