Abstract
We introduce a duality for affine iterated function systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson’s canonical measure μ, and we ask when μ is a spectral measure, i.e., when the Hilbert space \({L^2( \mu)}\) has an orthonormal basis (ONB) of exponentials \(\{{e_\lambda, | \lambda \in \Lambda}\}\) . We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra for X. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for X. Our results in turn are illustrated with concrete Sierpinski like fractals in dimensions 2 and 3.
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Andersson, S.I., Lapidus, M.L.: Spectral geometry: an introduction and background material for this volume. In: Andersson, S.I., Lapidus, M.L. (eds.) Progress in Inverse Spectral Geometry, pp. 1–14. Trends in Mathematics, Birkhäuser Verlag, Basel (1997)
Bandt, C., Hung, N.V., Rao, H.: On the open set condition for self-similar fractals. Proc. Am. Math. Soc. 134(5), 1369–1374 (electronic) (2006)
Barnsley, M.F., Devaney, R.L., Mandelbrot, B.B., Peitgen, H.O., Saupe, D., Voss, R.F.: The Science of Fractal Images. Springer, New York (1988). With contributions by Yuval Fisher and Michael McGuire
D’Andrea, J., Merrill, K.D., Packer, J.: Fractal wavelets of Dutkay-Jorgensen type for the Sierpinski gasket spaces. University of Colorado (preprint) (2006)
Dutkay D.E. and Jorgensen P.E.T. (2006). Iterated function systems, Ruelle operators and invariant projective measures. Math. Comp. 75: 1931–1970
Dutkay D.E. and Jorgensen P.E.T. (2006). Wavelets on fractals. Rev. Mat. Iberoamericana 22: 131–180
Dutkay, D.E., Jorgensen, P.E.T.: Harmonic analysis and dynamics for affine iterated function systems. Houston J. Math., to appear, http://arxiv.org/abs/math.DS/0502277
Dutkay, D.E., Jorgensen, P.E.T.: Fourier frequencies in affine iterated function systems. Preprint (2006)
Falconer K. (1997). Techniques in Fractal Geometry. Wiley, Chichester
Farkas, B., Matolcsi, M., Mora, P.: On Fuglede’s conjecture and the existence of universal spectra. Preprint http://arxiv.org/abs/math.CA/0612016 (2006)
Fuglede B. (1974). Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16: 101–121
Haagerup, U.: Orthogonal maximal abelian *-subalgebras of the n × n matrices and cyclic n-roots. In: Doplicher, S., Longo, R., Roberts, J.E., Zsido, L. (eds.) Operator Algebras and Quantum Field Theory (Rome, 1996), pp. 296–322. International Press, Cambridge (1997)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, Vol. 2: Structure and Analysis for Compact Groups: Analysis on Locally Compact Abelian Groups. Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer, New York (1970)
Hutchinson J.E. (1981). Fractals and self-similarity. Indiana Univ. Math. J. 30(5): 713–747
Iosevich A., Katz N. and Pedersen S. (1999). Fourier bases and a distance problem of Erdős. Math. Res. Lett. 6(2): 251–255
Iosevich A. and Pedersen S. (1998). Spectral and tiling properties of the unit cube. Internat. Math. Res. Notices 1998(16): 819–828
Jorgensen P.E.T. (1982). Spectral theory of finite volume domains in R n. Adv. Math. 44(2): 105–120
Jorgensen P.E.T. and Pedersen S. (1987). Harmonic analysis on tori. Acta Appl. Math. 10(1): 87–99
Jorgensen, P.E.T., Pedersen, S: An algebraic spectral problem for \(L^{2}(\Omega),\Omega\subset{\bf R}^n\), C. R. Acad. Sci. Paris Sér. I Math. 312(7):495–498
Jorgensen P.E.T. and Pedersen S. (1992). Spectral theory for Borel sets in R n of finite measure. J. Funct. Anal. 107(1): 72–104
Jorgensen P.E.T. and Pedersen S. (1993). Group-theoretic and geometric properties of multivariable Fourier series. Exposition. Math. 11(4): 309–329
Jorgensen P.E.T. and Pedersen S. (1998). Dense analytic subspaces in fractal L 2-spaces. J. Anal. Math. 75: 185–228
Kahane, J.-P.: Géza Freud and lacunary Fourier series. J. Approx. Theory 46(1), 51–57 (1986). Papers dedicated to the memory of Géza Freud
Kigami J. (2004). Local Nash inequality and inhomogeneity of heat kernels. Proc. Lond. Math. Soc. (3) 89(2): 525–544
Kigami J. and Lapidus M.L. (2001). Self-similarity of volume measures for Laplacians on p.c.f. self-similar fractals. Comm. Math. Phys. 217(1): 165–180
Kigami J., Strichartz R.S. and Walker K.C. (2001). Constructing a Laplacian on the diamond fractal. Exp. Math. 10(3): 437–448
Kolountzakis M.N. and Matolcsi M. (2006). Tiles with no spectra. Forum Math. 18(3): 519–528
Łaba I. and Wang Y. (2006). Some properties of spectral measures. Appl. Comput. Harmon. Anal. 20(1): 149–157
Lagarias J.C. and Wang Y. (1996). Self-affine tiles in R n. Adv. Math. 121(1): 21–49
Lagarias J.C. and Wang Y. (1997). Integral self-affine tiles in R n, II: Lattice tilings. J. Fourier Anal. Appl. 3(1): 83–102
Lam T.Y. and Leung K.H. (2000). On vanishing sums of roots of unity. J. Algebra 224(1): 91–109
Lapidus M.L., Neuberger J.W., Renka R.J. and Griffith C.A. (1996). Snowflake harmonics and computer graphics: numerical computation of spectra on fractal drums. Internat. J. Bifur. Chaos Appl. Sci. Eng. 6(7): 1185–1210
Matolcsi, M.: Fuglede’s conjecture fails in dimension 4. Preprint http://arxiv.org/abs/math. CA/0611936 (2006)
Pedersen S. and Wang Y. (2001). Universal spectra, universal tiling sets and the spectral set conjecture. Math. Scand. 88(2): 246–256
Senechal M. (1995). Quasicrystals and Geometry. Cambridge University Press, Cambridge
Sierpiński, W.: General topology. Mathematical expositions, No. 7, University of Toronto Press, Toronto (1952). Translated by C. Cecilia Krieger
Stewart I. (1995). Four encounters with Sierpiński’s gasket. Math. Intelligencer 17(1): 52–64
Strichartz R.S. and Wang Y. (1999). Geometry of self-affine tiles, I. Indiana Univ. Math. J. 48(1): 1–23
Tao T. (2004). Fuglede’s conjecture is false in 5 and higher dimensions. Math. Res. Lett. 11(2-3): 251–258
Tao T. and Vu V. (2006). On random ±1 matrices: singularity and determinant. Random Struct. Algorithms 28(1): 1–23
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Research supported in part by the National Science Foundation DMS 0457491.
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Dutkay, D.E., Jorgensen, P.E.T. Analysis of orthogonality and of orbits in affine iterated function systems. Math. Z. 256, 801–823 (2007). https://doi.org/10.1007/s00209-007-0104-9
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DOI: https://doi.org/10.1007/s00209-007-0104-9