Abstract
We consider representations of Cuntz–Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron–Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Albeverio, S., Kozyrev, S.V.: Multidimensional basis of p-adic wavelets and representation theory. arXiv:0903.0461
Bodin M.: Wavelets and Besov spaces on Mauldin–Williams fractals. Real Anal. Exch. 32(1), 119–144 (2006)
Bratteli O., Jorgensen P.E.T.: Iterated function systems and permutation representations of the Cuntz algebra. Mem. Am. Math. Soc. 139, 663 (1999)
Bratteli, O., Jorgensen, P.E.T., Ostrowsky, V.: Representation theory and numerical AF-invariants. The representations and centralizers of certain states on O d . Mem. Am. Math. Soc. 168, xviii+178pp (2004)
Consani C., Marcolli M.: Noncommutative geometry, dynamics, and ∞-adic Arakelov geometry. Selecta Math. (N.S.) 10(2), 167–251 (2004)
Consani C., Marcolli M.: Spectral triples from Mumford curves. Int. Math. Res. Not. 36, 1945–1972 (2003)
Cornelissen, G., Marcolli, M., Reihani, K., Vdovina, A.: Noncommutative geometry on trees and buildings. In: Traces in Number Theory, Geometry and Quantum Fields. Aspects of Mathematics, vol. E38, pp. 73–98. Vieweg, Braunschweig (2008)
Crovella, M., Kolaczyk, E.: Graph wavelets for spatial traffic analysis. In: Proceedings of IEEE Infocom 2003, San Francisco, CA, USA, April (2003)
Cuntz J., Krieger W.: A class of C*-algebras and topological Markov chains. Invent. Math. 56(3), 251–268 (1980)
Davidson K.R., Pitts D.R.: Invariant subspaces and hyper-reflexivity for free semigroups algebras. Proc. Lond. Math. Soc. 78, 401–430 (1999)
Daubechies, I.: Ten lectures on wavelets. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61. Society for Industrial and Applied Mathematics, Philadelphia (1992)
Dutkay D.E., Jorgensen P.E.T.: Iterated function systems, Ruelle operators, and invariant projective measures. Math. Comp. 75, 1931–1970 (2006)
Dutkay, D.E., Jorgensen, P.E.T.: Methods from multiscale theory and wavelets applied to nonlinear dynamics. In: Wavelets, Multiscale Systems and Hypercomplex Analysis. Operator Theory: Advances and Applications vol. 167, pp. 87–126. Birkhäuser, Basel (2006)
Dutkay D.E., Jorgensen P.E.T.: Wavelets on fractals. Rev. Mat. Iberoam. 22(1), 131–180 (2006)
Jonsson A.: Wavelets on fractals and Besov spaces. J. Fourier Anal. Appl. 4(3), 329–340 (1998)
Jorgensen P.E.T.: Iterated function systems, representations, and Hilbert space. Int. J. Math. 15(8), 813–832 (2004)
Jorgensen, P.E.T.: Use of operator algebras in the analysis of measures from wavelets and iterated function systems. In: Operator Theory, Operator Algebras, and Applications. Contemporary Mathematics, vol. 414, pp. 13–26. American Mathematical Society, Providence (2006)
Jorgensen P.E.T.: Measures in wavelet decompositions. Adv. Appl. Math. 34(3), 561–590 (2005)
Jorgensen P.E.T., Song M.S.: Optimal decompositions of translations of L 2-functions. Complex Anal. Oper. Theory 2(3), 449–478 (2008)
Kawamura K.: The Perron–Frobenius operators, invariant measures and representations of the Cuntz–Krieger algebra. J. Math. Phys. 46, 083514 (2005)
Kawamura, K.: Permutative representations of the Cuntz–Krieger algebras. arXiv:math/0508273
Kesseböhmer M., Stadlbauer M., Stratmann B.O.: Lyapunov spectra for KMS states on Cuntz–Krieger algebras. Math. Z. 256, 871–893 (2007)
Manin Yu.I., Marcolli M.: Continued fractions, modular symbols, and noncommutative geometry. Selecta Math. (New Ser.) 8(3), 475–521 (2002)
Marcolli M.: Limiting modular symbols and the Lyapunov spectrum. J. Number Theory 98(2), 348–376 (2003)
Marcolli, M.: Modular curves, C*-algebras, and chaotic cosmology. In: Frontiers in Number Theory, Physics and Geometry, II, pp. 361–372. Springer, Berlin (2007)
Strichartz R.S.: Wavelet expansions of fractal measures. J. Geom. Anal. 1(3), 269–289 (1991)
Strichartz R.S.: Wavelets and self-affine tilings. Constr. Approx. 9(2–3), 327–346 (1993)
Strichartz R.S.: Piecewise linear wavelets on Sierpinski gasket type fractals. J. Fourier Anal. Appl. 3(4), 387–416 (1997)
Acknowledgments
Part of this work was done during a stay of the authors at the Max Planck Institute for Mathematics, which we thank for the hospitality and support. M. Marcolli was partially supported by NSF grant DMS-0651925.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Palle Jorgensen.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Marcolli, M., Paolucci, A.M. Cuntz–Krieger Algebras and Wavelets on Fractals. Complex Anal. Oper. Theory 5, 41–81 (2011). https://doi.org/10.1007/s11785-009-0044-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-009-0044-y