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Ergodic Theory of Kusuoka Measures
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Mathematics Institute, University of Warwick, Coventry, UK.
2017 (English)In: Journal of Fractal Geometry, ISSN 2308-1317, Vol. 4, no 2, 185-214 p.Article in journal (Refereed) Published
Abstract [en]

In the analysis on self-similar fractal sets, the Kusuoka measure plays an important role. Here we investigate the Kusuoka measure from an ergodic theoretic viewpoint, seen as an invariant measure on a symbolic space. Our investigation shows that the Kusuoka measure generalizes Bernoulli measures and their properties to higher dimensions of an underlying finite dimensional vector space. Our main result is that the transfer operator on functions has a spectral gap when restricted to a certain Banach space that contains the Hölder continuous functions, as well as the highly discontinuous g" role="presentation">g-function associated to the Kusuoka measure. As a consequence, we obtain exponential decay of correlations. In addition, we provide some explicit rates of convergence for a family of generalized Sierpinski gaskets.

Place, publisher, year, edition, pages
European Mathematical Society Publishing House, 2017. Vol. 4, no 2, 185-214 p.
Keyword [en]
Kusuoka measure, energy Laplacian, transfer operator, quasi-compactness, g-measure
National Category
Geometry
Identifiers
URN: urn:nbn:se:hig:diva-24121DOI: 10.4171/JFG/49OAI: oai:DiVA.org:hig-24121DiVA: diva2:1107559
Available from: 2017-06-09 Created: 2017-06-09 Last updated: 2017-07-05Bibliographically approved

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Citation style
  • apa
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  • ieee
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  • Other locale
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Output format
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