Book contents
- Frontmatter
- Contents
- Preface
- Conference Photographs
- 1 Growth of Groups and Wreath Products
- 2 Random Walks on Some Countable Groups
- 3 The Cost of Distinguishing Graphs
- 4 A Construction of the Measurable Poisson Boundary: From Discrete to Continuous Groups
- 5 Structure Trees, Networks and Almost Invariant Sets
- 6 Amenability of Trees
- 7 Group-Walk Random Graphs
- 8 Ends of Branching Random Walks on Planar Hyperbolic Cayley Graphs
- 9 Amenability and Ergodic Properties of Topological Groups: From Bogolyubov Onwards
- 10 Schreier Graphs of Grigorchuk's Group and a Subshift Associated to a Nonprimitive Substitution
- 11 Thompson's Group F is Not Liouville
- 12 A Proof of the Subadditive Ergodic Theorem
- 13 Boundaries of Zn-Free Groups
- 14 Buildings, Groups of Lie Type and Random Walks
- 15 On Some Random Walks Driven by Spread-Out Measures
- 16 Topics on Mathematical Crystallography
- References
10 - Schreier Graphs of Grigorchuk's Group and a Subshift Associated to a Nonprimitive Substitution
Published online by Cambridge University Press: 20 July 2017
Book contents
- Frontmatter
- Contents
- Preface
- Conference Photographs
- 1 Growth of Groups and Wreath Products
- 2 Random Walks on Some Countable Groups
- 3 The Cost of Distinguishing Graphs
- 4 A Construction of the Measurable Poisson Boundary: From Discrete to Continuous Groups
- 5 Structure Trees, Networks and Almost Invariant Sets
- 6 Amenability of Trees
- 7 Group-Walk Random Graphs
- 8 Ends of Branching Random Walks on Planar Hyperbolic Cayley Graphs
- 9 Amenability and Ergodic Properties of Topological Groups: From Bogolyubov Onwards
- 10 Schreier Graphs of Grigorchuk's Group and a Subshift Associated to a Nonprimitive Substitution
- 11 Thompson's Group F is Not Liouville
- 12 A Proof of the Subadditive Ergodic Theorem
- 13 Boundaries of Zn-Free Groups
- 14 Buildings, Groups of Lie Type and Random Walks
- 15 On Some Random Walks Driven by Spread-Out Measures
- 16 Topics on Mathematical Crystallography
- References
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- Groups, Graphs and Random Walks , pp. 250 - 299Publisher: Cambridge University PressPrint publication year: 2017
References
[1] , , and : Benjamini-Schramm convergence and pointwise convergence of the spectral measure, Preprint (2013).
[2] and : Automatic Sequences: Theory, Applications, Generalization, Cambridge University Press, (2003).
[3] : Minimal Flows and their extensions, North-Holland Mathematical Studies 153, Elsevier (1988).
[4] , , , and : Equicontinuous factors, proximality and Ellis semigroup for Delone sets, to appear in [53].
[5] and : Aperiodic Order: Volume 1, A Mathematical Invitation, Encyclopedia of Mathematics and its Applications 149, Cambridge University Press, (2014).
[6] , , and : A characterization of model sets by dynamical systems, Ergodic Theory Dynam. Systems 27 (2007), 341–82.Google Scholar
[7] and : Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergod. Th. & Dynam. Systems 24 (2004) 1867–93.Google Scholar
[8] and (eds.): Directions in mathematical quasicrystals, CRM Monogr. Ser. 13, Amer. Math. Soc., Providence, RI (2000).
[9] and : Proximality and pure point spectrum for tiling dynamical systems, Michigan Journal of Mathematics 62 (2013), 793–822.Google Scholar
[10] and : On the spectrum of Hecke type operators related to some fractal groups, Tr. Mat. Inst. Steklova 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, 5–45; translation in Proc. Steklov Inst. Math. (2000), no. 4 (231), 1–41.Google Scholar
[11] , , and : From fractal groups to fractal sets in: “Fractals in Graz” (P. Grabner and W. Woess, eds.), Trends in Mathematics, Birkäuser Verlag, Basel, (2003), 25–118.
[12] and : Spectrum of Lebesgue measure zero for Jacobi matrices of quasicrystals, Math. Phys. Anal. Geom. 16 (2013), 289–308.Google Scholar
[13] and : Continuity of the spectrum of a field of selfadjoint operators, arXiv:1507.04641.
[14] , , and : Delone sets with finite local complexity: Linear repetitivity versus positivity of weights, Disc. Comput. Geom. 49 (2013), 335–47.Google Scholar
[15] : Topological full groups of minimal subshifts with subgroups of intermediate growth, Preprint 2014 (arXiv:1408.0762).
[16] : A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J. 52 (1985), 723–52.Google Scholar
[17] and : Liapunov stability and adding machines, Ergodic Theory Dynam. Systems 15 (1995), 271–90.Google Scholar
[18] , ,, and : Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer, Berlin, 1987.
[19] and : Spectral theory of random Schrödinger operators, Birkhäuser Boston, Boston, MA, (1990).
[20] : Singular continuous spectrum for a class of substitution Hamiltonians, Lett. Math. Phys. 46 (1998), 303–11.Google Scholar
[21] : Gordon-type arguments in the spectral theory of onedimensional quasicrystals, in [8], 277–305.
[22] , , and : Spectral properties of Schrödinger operators arising in the study of quasicrystals, to appear in [53].
[25] and : A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem, Duke Math. J. 133 (2006), 95–123.Google Scholar
[26] and : Substitution dynamical systems: characterization of linear repetitivity and applications, J. Math. Anal. Appl. 321 (2006), 766–80.Google Scholar
[27] and : Palindrome complexity bounds for primitive substitution sequences, Disc. Math. 222 (2000), 259–67.Google Scholar
[28] , , , and : Schreier graphs of the Basilica group, J. Mod. Dyn. 4 (2010), 167–205.Google Scholar
[29] : The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), 221–39.
[30] : Linearly recurrent subshifts have a finite number of nonperiodic subshift factors, Ergod. Th. & Dynam. Sys. 20 (2000), 1061–78.Google Scholar
[31] , private communication, 2015.
[32] , , and : Substitution dynamical systems, Bratteli diagrams and dimension groups, Ergod. Th. & Dynam. Sys. 19 (1999), 953–93.Google Scholar
[33] , private communication.
[34] , , and : Full groups of Cantor minimal systems, Israel J. Math. 111 (1999), 285–320.Google Scholar
[36] : On Burnside's problem on periodic groups. (Russian) Funktsional. Anal. i Prilozhen. 14 (1980), 53–4.Google Scholar
[37] : Degrees of growth of finitely generated groups and the theory of invariant means (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), 939–85.Google Scholar
[38] : Some problems of the dynamics of group actions on rooted trees (Russian), Tr. Mat. Inst. Steklova 273 (2011), Sovremennye Problemy Matematiki, 72–191; translation in Proc. Steklov Inst. Math. 273 (2011), 64–175.Google Scholar
[39] , , and : Spectra of Schreier graphs of Grigorchuk's group and Schrödinger operators with aperiodic order, arXiv:1412.6822.
[40] , , , Combinatorics of the subshift associated to Grigorchuk's group, to appear in: Proc. Steklov Inst. Math.
[41] , , , and : Selfsimilar groups, automatic sequences, and unitriangular representations, arXiv:1409.5027.
[42] and : Self-similar groups, operator algebras and Schur complements, Journal of Modern Dynamics, 1, (2007) 323–70.Google Scholar
[43] , , and : Automata, dynamical systems and infinite groups, Proc. Steklov Inst. Math. 231 (2000), 134–214.Google Scholar
[44] and : Asymptotic aspects of Schreier graphs and Hanoi Towers groups, Comptes Rendus Math. Acad. 342 (2006), 545–50.Google Scholar
[45] and : Schreier spectrum of the Hanoi towers group on three pegs, Proceedings of Symposia in Pure Mathematics, 77 (2008), 183–98.Google Scholar
[46] and : The lamplighter group as a group generated by a 2-state automaton, and its spectrum, Geom. Dedicata 87 (2001), 209–44.Google Scholar
[47] and . The Ihara Zeta function of infinite graphs, the KNS spectral measure and integrable maps, in: Random walks and geometry, V. Kaimanovich (eds.), (0), 141–80.Google Scholar
[48] : Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10 (1989), 95–100.Google Scholar
[49] : On an estimate concerning quantum diffusion in the presence of a fractal spectrum, Europhys. Lett. 21 (1993), 725–33.Google Scholar
[50] and : Lower bounds on wave packet propagation by packing dimensions of spectral measures, Math. Phys. Electron. J. 5 (1999), Paper 1, 16 pp.Google Scholar
[51] and , Matrix Analysis, Cambridge University Press, (1985).
[52] and : Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2) 178 (2013), 775–87.Google Scholar
[53] , , and (eds.): Directions in aperiodic order, to appear in: Progress in Mathematics, Birkhäuser.
[54] , , : Localization problem in one dimension: mapping and escape, Phys. Rev. Lett. 50 (1983), 1870–2.Google Scholar
[55] : Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys. 1 (1989), 129–33.Google Scholar
[56] and : Repetitive Delone sets and quasicrystals, Ergod. Th. & Dynam. Sys. 23 (2003), 831–67.Google Scholar
[57] : Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), 406–45.Google Scholar
[58] , , and : Pure point dynamical and diffraction spectra, Annales Henri Poincaré 3 (2002) 1003–18.Google Scholar
[59] : Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals, Comm. Math. Phys. 227 (2002), 119–30.Google Scholar
[60] : Random operators and crossed products, Math. Phys. Anal. Geom. 2 (1999), 197–220.Google Scholar
[61] :Uniform ergodic theorems on subshifts over a finite alphabet, Ergod. Th. & Dynam. Sys. 22 (2002), 245–55.Google Scholar
[62] and : Stationary processes with pure point diffraction, preprint 2012.
[63] , , and : Groupoids, von Neumann algebras and the integrated density of states, Math. Phys. Anal. Geom. 10 (2007), 1–41.Google Scholar
[64] , : Pure point spectrum for measure dynamical systems on locally compact Abelian groups, J.Math. Pures Appl. 92 (2009) 323–341.Google Scholar
[65] and : An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, (1995).
[66] : A set of defining relations for the Grigorchuk group (Russian), Mat. Zametki 38 (1985), 503–16. English translation in: Math. Notes 38 (1985), 784–92.Google Scholar
[67] : Topological full groups of minimal subshifts with subgroups of intermediate growth, Preprint 2014 (arXiv:1408.0762).
[68] : Some remarks on topological full groups of Cantor minimal systems, Internat. J. Math. 17 (2006), 231–51.Google Scholar
[69] (ed): The Mathematics of Long-Range Aperiodic Order, NATO-ASI Series C 489, Kluwer, Dordrecht (1997) 239–68.
[70] : Self-similar groups, Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005.
[71] , , , , and : Onedimensional Schrödinger equation with an almost periodic potential, Phys. Rev. Lett. 50 (1983), 1873–7.Google Scholar
[72] : Substitution dynamical systems–spectral analysis, Lecture Notes in Mathematics, 1294. Springer-Verlag, Berlin, 1987.
[73] : Quasicrystals and Geometry, Cambridge University Press, (1995).
[74] , , , and : Metallic phase with long-range orientational order and no translation symmetry, Phys. Rev. Lett. 53 (1984) 183–5.Google Scholar
[75] : Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discr. Comput. Geom. 20 (1998), 265–79.Google Scholar
[76] : Hausdorff dimension in a family of self-similar groups, Geometriae Dedicata 124 (2007), 213–36.Google Scholar
[77] : Schrödinger difference equation with deterministic ergodic potentials, in Beyond Quasicrystals (Les Houches, 1994), Springer, Berlin (1995), 481–549.
[78] , Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs 72, Amer. Math. Soc. Providence (2000).
[79] : On a substitution subshift related to the Grigorchuk group, preprint 2009, (ArXiv:0910.4800).
[80] : Notes on the Schreier graphs of the Grigorchuk group, Dynamical systems and group actions, 221–248, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, (2012).
[81] , An Introduction to Ergodic Theory, Springer, New York (1982).
[82] : Linear operators in Hilbert spaces, Graduate Texts in Mathematics 68, Springer-Verlag, New York-Berlin (1980).
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