Spectral geometries on a compact metric space
HTML articles powered by AMS MathViewer
- by
S. Buyalo
Translated by: the author - St. Petersburg Math. J. 30 (2019), 821-839
- DOI: https://doi.org/10.1090/spmj/1571
- Published electronically: July 26, 2019
- PDF | Request permission
Abstract:
The notion of a spectral geometry on a compact metric space $X$ is introduced. This is a sort of discrete approximation of $X$ motivated by the notion of a spectral triple from noncommutative geometry. A set of axioms characterizing spectral geometries is given. Bounded deformations of spectral geometries are studied and the relationship between the dimension of a spectral geometry and more traditional dimensions of metric spaces is investigated.References
- G. Bouligand, Ensembles impropes et nombre dimensionnel, Bull. Sci. Math. (2) 52 (1928) 320–344 & 361–376.
- S. V. Buyalo, Measurability of self-similar spectral geometries, Algebra i Analiz 12 (2000), no. 3, 1–39 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 3, 353–377. MR 1778189
- Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
- Jacques Dixmier, Existence de traces non normales, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A1107–A1108 (French). MR 196508
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
- Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
- A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in function spaces, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 3–86 (Russian). MR 0112032
- M. A. Naĭmark, Normed rings, P. Noordhoff N. V., Groningen, 1959. Translated from the first Russian edition by Leo F. Boron. MR 0110956
- Claude Tricot Jr., Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 1, 57–74. MR 633256, DOI 10.1017/S0305004100059119
Bibliographic Information
- S. Buyalo
- Affiliation: St. Petersburg Branch Steklov Mathematical Institute, 27 Fontanka, 191023 St. Petersburg, Russia
- Email: sbuyalo@pdmi.ras.ru
- Received by editor(s): June 1, 2018
- Published electronically: July 26, 2019
- Additional Notes: This work is supported by the Program of the Presidium of the Russian Academy of Sciences no. 01 “Fundamental Mathematics and its Applications” under Grant PRAS-18-01
- © Copyright 2019 American Mathematical Society
- Journal: St. Petersburg Math. J. 30 (2019), 821-839
- MSC (2010): Primary 51K10
- DOI: https://doi.org/10.1090/spmj/1571
- MathSciNet review: 3856102