Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-03T01:17:53.645Z Has data issue: false hasContentIssue false
  • English
  • Français

Ergodic Properties of Randomly Coloured Point Sets

Published online by Cambridge University Press:  20 November 2018

Peter Müller  and
Christoph Richard
Peter Müller
Affiliation:
Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraβe 39, 80333 München, Germany, e-mail: [email protected]
Christoph Richard
Affiliation:
Department für Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraβe 11, 91058 Erlangen, Germany, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide a framework for studying randomly coloured point sets in a locally compact second-countable space on which a metrizable unimodular group acts continuously and properly. We first construct and describe an appropriate dynamical system for uniformly discrete uncoloured point sets. For point sets of finite local complexity, we characterize ergodicity geometrically in terms of pattern frequencies. The general framework allows us to incorporate a random colouring of the point sets. We derive an ergodic theorem for randomly coloured point sets with finite-range dependencies. Special attention is paid to the exclusion of exceptional instances for uniquely ergodic systems. The setup allows for a straightforward application to randomly coloured graphs

Keywords

37B5037A30Delone setsdynamical systems
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 2 , 01 April 2013 , pp. 349 - 402
Copyright
Copyright © Canadian Mathematical Society 2013

References

[AMN] Abels, H., Manoussos, A., and Noskov, G., Proper actions and proper invariant metrics. J. London Math. Soc. 83(2011), no. 3, 619–636. http://dx.doi.org/10.1112/jlms/jdq091 Google Scholar
[A] Adler, R. J., The geometry of random fields. Wiley Series in Probability and Mathematical Statistics, JohnWiley & Sons, Ltd., Chichester, 1981.Google Scholar
[AI] Akama, Y. and S. Iizuka, , Random fields on model sets with localized dependency and their diffraction. arxiv:1107.4548 Google Scholar
[BBM] Baake, M., Birkner, M., and Moody, R. V., Diffraction of stochastic point sets: explicitly computable examples. Commun. Math. Phys. 293(2010), no. 3, 611–660. http://dx.doi.org/10.1007/s00220-009-0942-x Google Scholar
[BL] Baake, M. and Lenz, D., Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra. Ergodic Theory Dynam. Systems 24(2004), no. 6, 1867–1893. http://dx.doi.org/10.1017/S0143385704000318 Google Scholar
[BLM] Baake, M., Lenz, D., and Moody, R. V., Characterization of model sets by dynamical systems. Ergodic Theory Dynam. Systems 27(2007), no. 2, 341–382. http://dx.doi.org/10.1017/S0143385706000800 Google Scholar
[BZ]] Baake, M. and Zint, N., Absence of singular continuous diffraction for multi-component lattice gas models. J. Stat. Phys. 130(2008), no. 4, 727–740. http://dx.doi.org/10.1007/s10955-007-9445-3 Google Scholar
[Bau] Bauer, H., Measure and integration theory. de Gruyter Studies in Mathematics, 26, Walter de Gruyter & Co., Berlin, 2001.Google Scholar
[BeM] Bekka, M. B. and Mayer, M., Ergodic theory and topological dynamics of group actions on homogeneous spaces. London Mathematical Society Lecture Note Series, 269, Cambridge University Press, Cambridge, 2000.Google Scholar
[BelHZ] Bellissard, J., Herrmann, D. J. L., and Zarrouati, M., Hulls of aperiodic solids and gap labelingtheorems. In: Directions in mathematical quasicrystals, CRM Monogr. Ser., 13, American Mathematical Society, Providence, RI, 2000, pp. 207–258.Google Scholar
[Bou1] Bourbaki, N., General topology, Chapters 1–4. Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998.Google Scholar
[Bou2] Bourbaki, N., General topology, Chapters 5–10. Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998.Google Scholar
[Ch] Chatard, J., Sur une généralisation du théorème de Birkhoff.. R. Acad. Sci. Paris Sér. A-B 275(1972), A1135–A1138.Google Scholar
[DL] Damanik, D. and Lenz, D., Linear repetitivity. I. Uniform subadditive ergodic theorems and applications. Discrete Comput. Geom. 26(2001), no. 3. 411–428. http://dx.doi.org/10.007/s00454-001-0033-z Google Scholar
[Di] Diestel, R., Graph theory. Third ed., Graduate Texts in Mathematics, 173, Springer-Verlag, Berlin, 2005.Google Scholar
[Do] Doukhan, P., Mixing. Properties and examples. Lecture Notes in Statistics, 85, Springer-Verlag, New York, 1994.Google Scholar
[Fr] Frettlüh, D., Substitution tilings with statistical circular symmetry. European J. Combin. 29(2008), 1881–1893. http://dx.doi.org/10.1016/j.ejc.2008.01.006 Google Scholar
[Fu] Furstenberg, H., Recurrence in ergodic theory and combinatorial number theory. M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1981.Google Scholar
[GrS] Grünbaum, B. and Shephard, G. C., Tilings and patterns. An Introduction. A Series of Books in the Mathematical Sciences,W. H. Freeman and Co., New York, 1989.Google Scholar
[HR] Hewitt, E. and Ross, K. A., Abstract harmonic analysis. Vol. I. Structure of topological groups, theory, group representations. Second ed., Grundlehren der Mathematischen Wissenschaften, 115, Springer-Verlag, Berlin-New York, 1979.Google Scholar
[Ho1] Hof, A., Some remarks on discrete aperiodic Schrödinger operators. J. Stat. Phys. 72(1993), 1353–1374. http://dx.doi.org/10.1007/BF01048190 Google Scholar
[Ho2] Hof, A., A remark on Schrödinger operators on aperiodic tilings. J. Stat. Phys. 81(1995), no. 3–4, 851–855. http://dx.doi.org/10.1007/BF02179262 Google Scholar
[Ho3] Hof, A., Percolation on Penrose tilings. Canad. Math. Bull. 41(1998), no. 2, 166–177. http://dx.doi.org/10.4153/CMB-1998-026-0 Google Scholar
[I] Itô, K., Introduction to probability theory. Cambridge University Press, Cambridge, 1984.Google Scholar
[J] Janzen, D., Følner nets for semidirect products of amenable groups. Canad. Math. Bull. 51(2008), no. 1, 60–66. http://dx.doi.org/10.4153/CMB-2008-008-7 Google Scholar
[Ka] Kallenberg, O., Foundations of modern probability. Second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002.Google Scholar
[Ke] Kelley, J. L., General topology. Graduate Texts in Mathematics, 27, Springer-Verlag, New York-Berlin, 1975.Google Scholar
[KM] Kirsch, W. and Müller, P., Spectral properties of the Laplacian on bond-percolation graphs. Math. Z. 252(2006), 899–916. http://dx.doi.org/10.1007/s00209-005-0895-5 Google Scholar
[KlLS] Klassert, S., Lenz, D., and Stollmann, P., Discontinuities of the integrated density of states for random operators on Delone sets. Commun. Math. Phys. 241(2003), no. 2–3, 235–243.Google Scholar
[Kr] Krengel, U., Ergodic theorems. With a supplement by Antoine Brunel, de Gruyter Studies in Mathematics, 6,Walter de Gruyter & Co., Berlin, 1985.Google Scholar
[KB] Kryloff, N. and Bogoliouboff, N. La théorie générale de la mesure dans son application àl’´etude des systèmes dynamiques de la mécanique non linéaire. Ann. of Math. 38(1937), no. 1, 65–113. http://dx.doi.org/10.2307/1968511 Google Scholar
[Kü1] Külske, C., Universal bounds on the selfaveraging of random diffraction measures. Probab. Theory Related Fields 126(2003), no. 1, 29–50. http://dx.doi.org/10.1007/s00440-003-0261-7 Google Scholar
[Kü2] Külske, C., Concentration inequalities for functions of Gibbs fields with application to diffraction and random Gibbs measures. Commun. Math. Phys. 239(2003), no. 1-2, 29–51. http://dx.doi.org/10.1007/s00220-003-0841-5 Google Scholar
[LP] Lagarias, J. C. and P. A. B., Pleasants, Repetitive Delone sets and quasicrystals. Ergodic Theory Dynam. Systems 23(2003), no. 3, 831–867. http://dx.doi.org/10.1017/S0143385702001566 Google Scholar
[LeMS] Lee, J.-Y.,Moody, R. V., and Solomyak, B., Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3(2002), no. 5, 1003–1018. http://dx.doi.org/10.1007/s00023-002-8646-1 Google Scholar
[LeS] Lee, J.-Y. and Solomyak, B., Pure point diffractive substitution Delone sets have the Meyer property. Discrete Comput. Geom. 39(2008), no. 13, 319–338. http://dx.doi.org/10.1007/s00454-008-9054-1 Google Scholar
[Len] Lenz, D., Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks. Commun. Math. Phys. 287(2009), no. 1, 225–258. http://dx.doi.org/10.1007/s00220-008-0594-2 Google Scholar
[LenRi] Lenz, D. and Richard, C., Pure point diffraction and cut and project schemes for measures: the smooth case. Math. Z. 256(2007), no. 2, 347–378. http://dx.doi.org/10.1007/s00209-006-0077-0 Google Scholar
[LenS1] Lenz, D. and Stollmann, P., Algebras of random operators associated to Delone dynamical systems. Math. Phys. Anal. Geom. 6(2003), no. 3, 269–290. http://dx.doi.org/10.1023/A:1024900532603Google Scholar
[LenS2] Lenz, D., Delone dynamical systems and associated random operators. In: Operator algebras and mathematical physics (Constanta 2001), Theta, Bucharest, 2003, pp. 267–285.Google Scholar
[LenS3] Lenz, D., An ergodic theorem for Delone dynamical systems and existence of the integrated density of states. J. Anal. Math. 97(2005), 1–24. http://dx.doi.org/10.1007/BF02807400 Google Scholar
[Lin] Lindenstrauss, E., Pointwise theorems for amenable groups. Invent. Math. 146(2001), no. 2, 259–295. http://dx.doi.org/10.1007/s002220100162 Google Scholar
[Ma] Matzutt, K. M., Diffraction of point sets with structural disorder. Ph.D. thesis, Universität Bielefeld, 2010.Google Scholar
[Mo] Moody, R. V., Model sets: a survey. In: From quasicrystals to more complex systems, Springer, Berlin, 2000, pp. 145–166.Google Scholar
[N] Nevo, A., Pointwise ergodic theorems for actions of groups. In: Handbook of dynamical systems, Vol. 1B, Elsevier, Amsterdam, 2006, pp. 871–982.Google Scholar
[O] Oxtoby, J. C., Ergodic sets. Bull. Amer. Math. Soc. 58(1952), 116–136. http://dx.doi.org/10.1090/S0002-9904-1952-09580-X Google Scholar
[P] Paterson, A. L. T., Amenability. Mathematical Surveys and Monographs, 29, American Mathematical Society, Providence, RI, 1988.Google Scholar
[Ra1] Radin, C., The pinwheel tilings of the plane. Ann. of Math. (2) 139(1994), no. 3, 661–702. http://dx.doi.org/10.2307/2118575 Google Scholar
[Ra2] Radin, C., Symmetry and tilings. Notices Amer. Math. Soc. 42(1995), no. 1, 26–31.Google Scholar
[Ra3] Radin, C., Aperiodic tilings, ergodic theory, and rotations. In: The mathematics of long-range aperiodic order (Waterloo, ON, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 489, Kluwer Acad. Publ., Dordrecht, 1997, pp. 499–519.Google Scholar
[RaW] Radin, C. and Wolff, M., Space tilings and local isomorphism. Geom. Dedicata 42(1992), no. 3, 355–360. http://dx.doi.org/10.1007/BF02414073 Google Scholar
[ReSi] Reed, M. and B.|Simon, Methods in modern mathematical physics. I. Functional analysis. Second ed., Academic Press, Inc., New York, 1980.Google Scholar
[RiHHB] Richard, C., Höffe, M., Hermisson, J., and Baake, M., Random tilings: concepts and examples. J. Phys. A 31(1998), 6385–6408. http://dx.doi.org/10.1088/0305-4470/31/30/007 Google Scholar
[Ru] Rudin, W., Functional analysis. Second ed., International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1991.Google Scholar
[S] Schlottmann, M., Generalized model sets and dynamical systems. In: Directions in mathematical quasicrystals, CRM Monogr. Ser., 13, American Mathematical Society, Providence, RI, 2000, pp. 143–159.Google Scholar
[Sl] Slawny, J., Ergodic properties of equilibrium states. Commun. Math. Phys. 80(1980), no. 4, 477–483. http://dx.doi.org/10.1007/BF01941658 Google Scholar
[So] Solomyak, B., Dynamics of self-similar tilings. Ergodic Theory Dynam. Systems 17(1997), no. 3, 695–738; Corrections to: “Dynamics of self–similar tilings”. Ergodic Theory Dynam. Systems 19(1999), no. 6, 1685. http://dx.doi.org/10.1017/S0143385797084988 Google Scholar
[SKM] Stoyan, D., Kendall, W. S., and Mecke, J., Stochastic geometry and its applications. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, JohnWiley & Sons, Chichester, 1987.Google Scholar
[St] Struble, R. A., Metrics in locally compact groups. Compositio Math. 28(1974), 217–222.Google Scholar
[Wa] Walters, P., An introduction to ergodic theory. Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.Google Scholar
[Wi] Wilson, B., Reiter nets for semidirect products of amenable groups and semigroups. Proc. Amer. Math. Soc. 137(2009), no. 11, 3823–3832. http://dx.doi.org/10.1090/S0002-9939-09-09957-2 Google Scholar
[Yo] Yokonuma, T., Discrete sets and associated dynamical systems in a non-commutative setting. Canad. Math. Bull. 48(2005), no. 2, 302–316. http://dx.doi.org/10.4153/CMB-2005-028-8 Google Scholar