Iterated function systems, Ruelle operators, and invariant projective measures
Authors:
Dorin Ervin Dutkay and Palle E. T. Jorgensen
Journal:
Math. Comp. 75 (2006), 1931-1970
MSC (2000):
Primary 28A80, 31C20, 37F20, 39B12, 41A63, 42C40, 47D07, 60G42, 60J45
DOI:
https://doi.org/10.1090/S0025-5718-06-01861-8
Published electronically:
May 31, 2006
MathSciNet review:
2240643
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space $X$ comes with a finite-to-one endomorphism $r\colon X\rightarrow X$ which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in $\mathbb {R}^d$, this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets $B, L$ in $\mathbb {R}^d$ of the same cardinality which generate complex Hadamard matrices.
Our harmonic analysis for these iterated function systems (IFS) $(X, \mu )$ is based on a Markov process on certain paths. The probabilities are determined by a weight function $W$ on $X$. From $W$ we define a transition operator $R_W$ acting on functions on $X$, and a corresponding class $H$ of continuous $R_W$-harmonic functions. The properties of the functions in $H$ are analyzed, and they determine the spectral theory of $L^2(\mu )$. For affine IFSs we establish orthogonal bases in $L^2(\mu )$. These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in $\mathbb {R}^d$.
- 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. MR 2036639, DOI https://doi.org/10.1007/s00365-003-0539-0
- 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. MR 2084397, DOI https://doi.org/10.1090/S0002-9947-04-03679-7
- 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. MR 1793194
- Krishna B. Athreya and Peter E. Ney, Branching processes, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 196. MR 0373040
- Stefan Bildea, Dorin Ervin Dutkay, and Gabriel Picioroaga, MRA super-wavelets, New York J. Math. 11 (2005), 1–19. MR 2154344
- Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089
- Ola Bratteli, Palle E. T. Jorgensen, and Geoffrey L. Price, Endomorphisms of ${\scr B}({\scr H})$, Quantization, nonlinear partial differential equations, and operator algebra (Cambridge, MA, 1994) Proc. Sympos. Pure Math., vol. 59, Amer. Math. Soc., Providence, RI, 1996, pp. 93–138. MR 1392986, DOI https://doi.org/10.1090/pspum/059/1392986
- Ola Bratteli and Palle E. T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 139 (1999), no. 663, x+89. MR 1469149, DOI https://doi.org/10.1090/memo/0663
- Ola Bratteli and Palle Jorgensen, Wavelets through a looking glass, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2002. The world of the spectrum. MR 1913212
- Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144 (1965). MR 194595, DOI https://doi.org/10.1007/BF02591353
- Xia Chen, Limit theorems for functionals of ergodic Markov chains with general state space, Mem. Amer. Math. Soc. 139 (1999), no. 664, xiv+203. MR 1491814, DOI https://doi.org/10.1090/memo/0664
- Jean-Pierre Conze and Albert Raugi, Fonctions harmoniques pour un opérateur de transition et applications, Bull. Soc. Math. France 118 (1990), no. 3, 273–310 (French, with English summary). MR 1078079
- 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. MR 1162107
- Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675
- D.E. Dutkay, P.E.T. Jorgensen, Wavelets on fractals, Rev. Mat. Iberoamericana, to appear, http://arXiv.org/abs/math.CA/0305443.
- D.E. Dutkay, P.E.T. Jorgensen, Martingales, endomorphisms, and covariant systems of operators in Hilbert space, preprint, 2004, http://arxiv.org/abs/math.CA/0407330 .
- D.E. Dutkay, P.E.T. Jorgensen, Disintegration of projective measures, Proc. Amer. Math. Soc., to appear, http://arxiv.org/abs/math.CA/0408151.
- Dorin Ervin Dutkay and Palle E. T. Jorgensen, Hilbert spaces of martingales supporting certain substitution-dynamical systems, Conform. Geom. Dyn. 9 (2005), 24–45. MR 2133804, DOI https://doi.org/10.1090/S1088-4173-05-00135-9
- Bent Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis 16 (1974), 101–121. MR 0470754, DOI https://doi.org/10.1016/0022-1236%2874%2990072-x
- 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. MR 1738093, DOI https://doi.org/10.1090/conm/247/03805
- Richard F. Gundy, Low-pass filters, martingales, and multiresolution analyses, Appl. Comput. Harmon. Anal. 9 (2000), no. 2, 204–219. MR 1777126, DOI https://doi.org/10.1006/acha.2000.0320
- Richard F. Gundy and Kazaros Kazarian, Stopping times and local convergence for spline wavelet expansions, SIAM J. Math. Anal. 31 (2000), no. 3, 561–573. MR 1741041, DOI https://doi.org/10.1137/S0036141097327392
- Aimo Hinkkanen, Gaven J. Martin, and Volker Mayer, Local dynamics of uniformly quasiregular mappings, Math. Scand. 95 (2004), no. 1, 80–100. MR 2091483, DOI https://doi.org/10.7146/math.scand.a-14450
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI https://doi.org/10.1512/iumj.1981.30.30055
- C. T. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classes d’opérations non complètement continues, Ann. of Math. (2) 52 (1950), 140–147 (French). MR 37469, DOI https://doi.org/10.2307/1969514
- Alex Iosevich and Steen Pedersen, Spectral and tiling properties of the unit cube, Internat. Math. Res. Notices 16 (1998), 819–828. MR 1643694, DOI https://doi.org/10.1155/S1073792898000506
- Palle E. T. Jørgensen, Spectral theory of finite volume domains in ${\bf R}^{n}$, Adv. in Math. 44 (1982), no. 2, 105–120. MR 658536, DOI https://doi.org/10.1016/0001-8708%2882%2990001-9
- P.E.T. Jorgensen, Analysis and Probability: Wavelets, Signals, Fractals, monograph manuscript, book to be published.
- Palle E. T. Jorgensen, Measures in wavelet decompositions, Adv. in Appl. Math. 34 (2005), no. 3, 561–590. MR 2123549, DOI https://doi.org/10.1016/j.aam.2004.11.002
- Palle E. T. Jorgensen and Steen Pedersen, Spectral theory for Borel sets in ${\bf R}^n$ of finite measure, J. Funct. Anal. 107 (1992), no. 1, 72–104. MR 1165867, DOI https://doi.org/10.1016/0022-1236%2892%2990101-N
- Palle E. T. Jorgensen and Steen Pedersen, Group-theoretic and geometric properties of multivariable Fourier series, Exposition. Math. 11 (1993), no. 4, 309–329. MR 1240363
- P. E. T. Jorgensen and S. Pedersen, Harmonic analysis of fractal measures, Constr. Approx. 12 (1996), no. 1, 1–30. MR 1389918, DOI https://doi.org/10.1007/s003659900001
- Palle E. T. Jorgensen and Steen Pedersen, Dense analytic subspaces in fractal $L^2$-spaces, J. Anal. Math. 75 (1998), 185–228. MR 1655831, DOI https://doi.org/10.1007/BF02788699
- Palle E. T. Jorgensen and Steen Pedersen, Spectral pairs in Cartesian coordinates, J. Fourier Anal. Appl. 5 (1999), no. 4, 285–302. MR 1700084, DOI https://doi.org/10.1007/BF01259371
- Shizuo Kakutani, On equivalence of infinite product measures, Ann. of Math. (2) 49 (1948), 214–224. MR 23331, DOI https://doi.org/10.2307/1969123
- Izabella Łaba and Yang Wang, On spectral Cantor measures, J. Funct. Anal. 193 (2002), no. 2, 409–420. MR 1929508, DOI https://doi.org/10.1006/jfan.2001.3941
- Jeffrey C. Lagarias, James A. Reeds, and Yang Wang, Orthonormal bases of exponentials for the $n$-cube, Duke Math. J. 103 (2000), no. 1, 25–37. MR 1758237, DOI https://doi.org/10.1215/S0012-7094-00-10312-2
- J. C. Lagarias and P. W. Shor, Cube-tilings of ${\bf R}^n$ and nonlinear codes, Discrete Comput. Geom. 11 (1994), no. 4, 359–391. MR 1273224, DOI https://doi.org/10.1007/BF02574014
- Jeffrey C. Lagarias and Yang Wang, Integral self-affine tiles in ${\bf R}^n$. II. Lattice tilings, J. Fourier Anal. Appl. 3 (1997), no. 1, 83–102. MR 1428817, DOI https://doi.org/10.1007/s00041-001-4051-2
- Jeffrey C. Lagarias and Yang Wang, Orthogonality criteria for compactly supported refinable functions and refinable function vectors, J. Fourier Anal. Appl. 6 (2000), no. 2, 153–170. MR 1754012, DOI https://doi.org/10.1007/BF02510658
- Ka-Sing Lau, Mang-Fai Ma, and Jianrong Wang, On some sharp regularity estimations of $L^2$-scaling functions, SIAM J. Math. Anal. 27 (1996), no. 3, 835–864. MR 1382836, DOI https://doi.org/10.1137/0527045
- Ka-Sing Lau, Jianrong Wang, and Cho-Ho Chu, Vector-valued Choquet-Deny theorem, renewal equation and self-similar measures, Studia Math. 117 (1995), no. 1, 1–28. MR 1367690, DOI https://doi.org/10.4064/sm-117-1-1-28
- Wayne M. Lawton, Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys. 32 (1991), no. 1, 57–61. MR 1083085, DOI https://doi.org/10.1063/1.529093
- Ricardo Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 27–43. MR 736567, DOI https://doi.org/10.1007/BF02584743
- R. Daniel Mauldin and Mariusz Urbański, Graph directed Markov systems, Cambridge Tracts in Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003. Geometry and dynamics of limit sets. MR 2003772
- 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. MR 1470912, DOI https://doi.org/10.1090/memo/0659
- S. Richter and R. F. Werner, Ergodicity of quantum cellular automata, J. Statist. Phys. 82 (1996), no. 3-4, 963–998. MR 1372433, DOI https://doi.org/10.1007/BF02179798
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
- David Ruelle, The thermodynamic formalism for expanding maps, Comm. Math. Phys. 125 (1989), no. 2, 239–262. MR 1016871
- Jennifer Seberry and Mieko Yamada, Hadamard matrices, sequences, and block designs, Contemporary design theory, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992, pp. 431–560. MR 1178508
- Boris Solomyak, On the random series $\sum \pm \lambda ^n$ (an Erdős problem), Ann. of Math. (2) 142 (1995), no. 3, 611–625. MR 1356783, DOI https://doi.org/10.2307/2118556
- Robert S. Strichartz, Remarks on: “Dense analytic subspaces in fractal $L^2$-spaces” [J. Anal. Math. 75 (1998), 185–228; MR1655831 (2000a:46045)] by P. E. T. Jorgensen and S. Pedersen, J. Anal. Math. 75 (1998), 229–231. MR 1655832, DOI https://doi.org/10.1007/BF02788700
- Robert S. Strichartz, Mock Fourier series and transforms associated with certain Cantor measures, J. Anal. Math. 81 (2000), 209–238. MR 1785282, DOI https://doi.org/10.1007/BF02788990
- R.S. Strichartz, Convergence of mock Fourier series, Duke Math. J., to appear.
- Peter Walters, Ruelle’s operator theorem and $g$-measures, Trans. Amer. Math. Soc. 214 (1975), 375–387. MR 412389, DOI https://doi.org/10.1090/S0002-9947-1975-0412389-8
- R.F. Werner, Unitary matrices with entries of equal modulus, preprint 1993, Universität Osnabruck.
- Masaya Yamaguti, Masayoshi Hata, and Jun Kigami, Mathematics of fractals, Translations of Mathematical Monographs, vol. 167, American Mathematical Society, Providence, RI, 1997. Translated from the 1993 Japanese original by Kiki Hudson. MR 1471705
- K\B{o}saku Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition. MR 1336382
Retrieve articles in Mathematics of Computation with MSC (2000): 28A80, 31C20, 37F20, 39B12, 41A63, 42C40, 47D07, 60G42, 60J45
Retrieve articles in all journals with MSC (2000): 28A80, 31C20, 37F20, 39B12, 41A63, 42C40, 47D07, 60G42, 60J45
Additional Information
Dorin Ervin Dutkay
Affiliation:
Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd, Piscataway, New Jersey 08854-8019
MR Author ID:
608228
Email:
ddutkay@math.rutgers.edu
Palle E. T. Jorgensen
Affiliation:
Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, Iowa 52242-1419
MR Author ID:
95800
ORCID:
0000-0003-2681-5753
Email:
jorgen@math.uiowa.edu
Keywords:
Measures,
projective limits,
transfer operator,
martingale,
fixed-point,
wavelet,
multiresolution,
fractal,
Hausdorff dimension,
Perron-Frobenius,
Julia set,
subshift,
orthogonal functions,
Fourier series,
Hadamard matrix,
tiling,
lattice,
harmonic function
Received by editor(s):
January 5, 2005
Received by editor(s) in revised form:
June 16, 2005
Published electronically:
May 31, 2006
Additional Notes:
This research was supported in part by the National Science Foundation DMS-0139473 (FRG)
Article copyright:
© Copyright 2006
American Mathematical Society