×

Hilbert spaces of martingales supporting certain substitution-dynamical systems. (English) Zbl 1128.37005

Summary: Let \(X\) be a compact Hausdorff space. We study finite-to-one mappings \(r\colon X\rightarrow X\), onto \(X\), and measures on the corresponding projective limit space \(X_\infty(r)\). We show that certain quasi-invariant measures on \(X_\infty(r)\) correspond in a one-to-one fashion to measures on \(X\) which satisfy two identities. Moreover, we identify those special measures on \(X_\infty(r)\) which are associated via our correspondence with a function \(V\) on \(X\), a Ruelle transfer operator \(R_V\), and an equilibrium measure \(\mu_V\) on \(X\).

MSC:

37A99 Ergodic theory
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
46G15 Functional analytic lifting theory
47D07 Markov semigroups and applications to diffusion processes
60G18 Self-similar stochastic processes
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Luigi Accardi, Alberto Frigerio, and John T. Lewis, Quantum stochastic processes, Publ. Res. Inst. Math. Sci. 18 (1982), no. 1, 97 – 133. · Zbl 0498.60099 · doi:10.2977/prims/1195184017
[2] Akram Aldroubi, David Larson, Wai-Shing Tang, and Eric Weber, Geometric aspects of frame representations of abelian groups, Trans. Amer. Math. Soc. 356 (2004), no. 12, 4767 – 4786. · Zbl 1054.43008 · doi:10.1090/S0002-9947-04-03679-7
[3] Jonathan Ashley, Brian Marcus, and Selim Tuncel, The classification of one-sided Markov chains, Ergodic Theory Dynam. Systems 17 (1997), no. 2, 269 – 295. · Zbl 0880.60067 · doi:10.1017/S0143385797069745
[4] Akram Aldroubi, Qiyu Sun, and Wai-Shing Tang, Nonuniform average sampling and reconstruction in multiply generated shift-invariant spaces, Constr. Approx. 20 (2004), no. 2, 173 – 189. · Zbl 1049.42017 · doi:10.1007/s00365-003-0539-0
[5] Larry Baggett, Alan Carey, William Moran, and Peter Ohring, General existence theorems for orthonormal wavelets, an abstract approach, Publ. Res. Inst. Math. Sci. 31 (1995), no. 1, 95 – 111. · Zbl 0849.42019 · doi:10.2977/prims/1195164793
[6] L. W. Baggett, P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer, An analogue of Bratteli-Jorgensen loop group actions for GMRA’s, Wavelets, frames and operator theory, Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 11 – 25. · Zbl 1073.42022 · doi:10.1090/conm/345/06238
[7] Lawrence W. Baggett and Kathy D. Merrill, Abstract harmonic analysis and wavelets in \?\(^{n}\), The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999) Contemp. Math., vol. 247, Amer. Math. Soc., Providence, RI, 1999, pp. 17 – 27. · Zbl 0957.42021 · doi:10.1090/conm/247/03795
[8] Viviane Baladi, Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, vol. 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. · Zbl 1012.37015
[9] Luis Barreira and Yakov B. Pesin, Lyapunov exponents and smooth ergodic theory, University Lecture Series, vol. 23, American Mathematical Society, Providence, RI, 2002. · Zbl 1195.37002
[10] Berndt Brenken and Palle E. T. Jorgensen, A family of dilation crossed product algebras, J. Operator Theory 25 (1991), no. 2, 299 – 308. · Zbl 0789.46056
[11] Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103 – 144 (1965). · Zbl 0127.03401
[12] Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. · Zbl 0776.42018
[13] Valentin Deaconu and Paul S. Muhly, \?*-algebras associated with branched coverings, Proc. Amer. Math. Soc. 129 (2001), no. 4, 1077 – 1086 (electronic). · Zbl 0971.46041 · doi:10.1090/S0002-9939-00-05697-5
[14] J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. · Zbl 0549.31001
[15] Dorin E. Dutkay, The wavelet Galerkin operator, J. Operator Theory 51 (2004), no. 1, 49 – 70. · Zbl 1061.42025
[16] D.E. Dutkay, P.E.T. Jorgensen, Wavelets on fractals, to appear in Rev. Mat. Iberoamericana. · Zbl 1104.42021
[17] D.E. Dutkay, P.E.T. Jorgensen, Martingales, endomorphisms, and covariant systems of operators in Hilbert space, preprint 2004, arxiv math.CA/0407330. · Zbl 1134.47305
[18] R. Gohm, B. Kummerer, T. Lang, Non-commutative symbolic coding, Preprint 2005, Greifswald University.
[19] Richard F. Gundy, Two remarks concerning wavelets: Cohen’s criterion for low-pass filters and Meyer’s theorem on linear independence, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999) Contemp. Math., vol. 247, Amer. Math. Soc., Providence, RI, 1999, pp. 249 – 258. · Zbl 0974.42024 · doi:10.1090/conm/247/03805
[20] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713 – 747. · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055
[21] Palle E. T. Jorgensen, Ruelle operators: functions which are harmonic with respect to a transfer operator, Mem. Amer. Math. Soc. 152 (2001), no. 720, viii+60. · Zbl 0995.46046 · doi:10.1090/memo/0720
[22] P.E.T. Jorgensen, Analysis and Probability, preprint 2004.
[23] P. E. T. Jorgensen and D. W. Kribs, Wavelet representations and Fock space on positive matrices, J. Funct. Anal. 197 (2003), no. 2, 526 – 559. · Zbl 1016.42024 · doi:10.1016/S0022-1236(02)00026-5
[24] Palle T. Jørgensen and Paul S. Muhly, Selfadjoint extensions satisfying the Weyl operator commutation relations, J. Analyse Math. 37 (1980), 46 – 99. · Zbl 0448.47012 · doi:10.1007/BF02797680
[25] Elias Katsoulis and David W. Kribs, Isomorphisms of algebras associated with directed graphs, Math. Ann. 330 (2004), no. 4, 709 – 728. · Zbl 1069.47072 · doi:10.1007/s00208-004-0566-6
[26] David W. Kribs, On bilateral weighted shifts in noncommutative multivariable operator theory, Indiana Univ. Math. J. 52 (2003), no. 6, 1595 – 1614. · Zbl 1048.46044 · doi:10.1512/iumj.2003.52.2375
[27] A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin-New York, 1977 (German). Reprint of the 1933 original. · Zbl 0007.21601
[28] Burkhard Kümmerer, Markov dilations on \?*-algebras, J. Funct. Anal. 63 (1985), no. 2, 139 – 177. · Zbl 0601.46062 · doi:10.1016/0022-1236(85)90084-9
[29] Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases of \?²(\?), Trans. Amer. Math. Soc. 315 (1989), no. 1, 69 – 87. · Zbl 0686.42018 · doi:10.1090/S0002-9947-1989-1008470-5
[30] Paul S. Muhly and Baruch Solel, Quantum Markov processes (correspondences and dilations), Internat. J. Math. 13 (2002), no. 8, 863 – 906. · Zbl 1057.46050 · doi:10.1142/S0129167X02001514
[31] J. Neveu, Discrete-parameter martingales, Revised edition, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed; North-Holland Mathematical Library, Vol. 10. · Zbl 0345.60026
[32] R. D. Nussbaum and S. M. Verduyn Lunel, Generalizations of the Perron-Frobenius theorem for nonlinear maps, Mem. Amer. Math. Soc. 138 (1999), no. 659, viii+98. · Zbl 0933.47036 · doi:10.1090/memo/0659
[33] Gelu Popescu, Commutant lifting, tensor algebras, and functional calculus, Proc. Edinb. Math. Soc. (2) 44 (2001), no. 2, 389 – 406. · Zbl 0986.47052 · doi:10.1017/S0013091598001059
[34] Gelu Popescu, Central intertwining lifting, suboptimization, and interpolation in several variables, J. Funct. Anal. 189 (2002), no. 1, 132 – 154. · Zbl 1013.46061 · doi:10.1006/jfan.2001.3861
[35] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. · Zbl 0925.00005
[36] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. · Zbl 0867.46001
[37] David Ruelle, The thermodynamic formalism for expanding maps, Comm. Math. Phys. 125 (1989), no. 2, 239 – 262. · Zbl 0702.58056
[38] David Ruelle, Application of hyperbolic dynamics to physics: some problems and conjectures, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 3, 275 – 278 (electronic). · Zbl 1102.37051 · doi:10.1090/S0273-0979-04-01023-7
[39] Meiyu Su, The information topology and true laminations for diffeomorphisms, Conform. Geom. Dyn. 8 (2004), 36 – 51 (electronic). · Zbl 1075.37006 · doi:10.1090/S1088-4173-04-00107-9
[40] Domokos Szász and Tamás Varjú, Local limit theorem for the Lorentz process and its recurrence in the plane, Ergodic Theory Dynam. Systems 24 (2004), no. 1, 257 – 278. · Zbl 1115.37009 · doi:10.1017/S0143385703000439
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.