×

The mapping class group of a minimal subshift. (English) Zbl 1467.37009

The mapping class group \(\mathcal{M}(T)\) of a Cantor minimal system \((X,T)\) is defined as the group of isotopy classes of the orientation-preserving homeomorphisms of the mapping torus \(\Sigma_T X\) of the \((X,T)\). Such a group measures symmetries of the suspension flow of the Cantor system. The authors establish a series of fundamental results for \(\mathcal{M}(T)\). More precisely, their Theorem 1.1 says that the \(\mathcal{M}(T)\) is an extension of \(\mathbf{Z}\) by a finite group. The second result (Theorem 1.2) establishes virtual commutativity of the mapping class group \(\mathcal{M}(T)\) provided some mild restriction on the Cantor system. Overall, the paper contains a trove of other interesting results, in particular connections to the AF \(C^*\)-algebras and their \(K\)-theory. The text is well motivated, clearly written and can be used by graduate students as an entry point to symbolic dynamics and \(C^*\)-algebra theory.

MSC:

37B10 Symbolic dynamics
37A55 Dynamical systems and the theory of \(C^*\)-algebras
46L85 Noncommutative topology
57K50 Low-dimensional manifolds of specific dimension 5 or higher
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] J. Aliste-Prieto and S. Petite,On the simplicity of homeomorphism groups of a tilable lamination, Monatsh. Math. 181 (2016), 285-300. · Zbl 1353.57029
[2] M. Barge and B. Diamond,A complete invariant for the topology of one-dimensional substitution tiling spaces, Ergodic Theory Dynam. Systems 21 (2001), 1333-1358. · Zbl 0986.37015
[3] M. Barge, B. Diamond, and C. Holton,Asymptotic orbits of primitive substitutions, Theoret. Computer Sci. 301 (2003), 439-450. · Zbl 1022.68107
[4] M. Barge and R. Swanson,Rigidity in one-dimensional tiling spaces, Topology Appl. 154 (2007), 3095-3099. · Zbl 1123.37010
[5] M. Barge and R. F. Williams,Classification of Denjoy continua, Topology Appl. 106 (2000), 77-89. · Zbl 0983.37013
[6] V. Berthé, H. Ei, S. Ito, and H. Rao,On substitution invariant Sturmian words: an application of Rauzy fractals, Theoret. Informatics Appl. 41 (2007), 329-349. · Zbl 1140.11014
[7] M. Boshernitzan,A unique ergodicity of minimal symbolic flows with linear block growth, J. Anal. Math. 44 (1984), 77-96. · Zbl 0602.28008
[8] M. Boyle,Flow equivalence of shifts of finite type via positive factorizations, Pacific J. Math. 204 (2002), 273-317. · Zbl 1056.37008
[9] M. Boyle, T. M. Carlsen, and S. Eilers,Flow equivalence and isotopy for subshifts, Dynam. Systems 32 (2017), 305-325. · Zbl 1381.37020
[10] M. Boyle, T. M. Carlsen, and S. Eilers,Corrigendum: “Flow equivalence and isotopy for subshifts”, Dynam. Systems 32 (2017), no. 3, ii. · Zbl 1381.37020
[11] M. Boyle and S. Chuysurichay,The mapping class group of a shift of finite type, J. Modern Dynam. 13 (2018), 115-145. · Zbl 1407.37020
[12] M. Boyle and D. Handelman,Orbit equivalence, flow equivalence and ordered cohomology, Israel J. Math. 95 (1996), 169-210. · Zbl 0871.58071
[13] M. Boyle, D. Lind, and D. Rudolph,The automorphism group of a shift of finite type, Trans. Amer. Math. Soc. 306 (1988), 71-114. · Zbl 0664.28006
[14] L. G. Brown, P. Green, and M. A. Rieffel,Stable isomorphism and strong Morita equivalence ofC∗-algebras, Pacific J. Math. 71 (1977), 349-363. · Zbl 0362.46043
[15] S. Chuysurichay,Positive rational strong shift equivalence and the mapping class group of a shift of finite type, thesis, Univ. of Maryland, College Park, MD, 2011.
[16] A. Clark and L. Sadun,When size matters: subshifts and their related tiling spaces, Ergodic Theory Dynam. Systems 23 (2003), 1043-1057. · Zbl 1042.37008
[17] E. M. Coven,Endomorphisms of substitution minimal sets, Z. Wahrsch. Verw. Gebiete 20 (1971), 129-133. · Zbl 0211.56405
[18] V. Cyr, J. Franks, B. Kra, and S. Petite,Distortion and the automorphism group of a shift, J. Modern Dynam. 13 (2018), 147-161. · Zbl 1407.37022
[19] V. Cyr and B. Kra,The automorphism group of a shift of linear growth: beyond transitivity, Forum Math. Sigma 3 (2015), art. e5, 27 pp. · Zbl 1321.37010
[20] V. Cyr and B. Kra,The automorphism group of a minimal shift of stretched exponential growth, J. Modern Dynam. 10 (2016), 483-495. · Zbl 1402.37013
[21] V. Cyr and B. Kra,The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc. 144 (2016), 613-621. · Zbl 1365.37019
[22] V. Cyr and B. Kra,Counting generic measures for a subshift of linear growth, J. Eur. Math. Soc. 21 (2019), 355-380. · Zbl 1437.37020
[23] S. Donoso, F. Durand, A. Maass, and S. Petite,On automorphism groups of low complexity subshifts, Ergodic Theory Dynam. Systems 36 (2016), 64-95. · Zbl 1354.37024
[24] F. Durand, N. Ormes, and S. Petite,Self-induced systems, J. Anal. Math. 135 (2018), 725-756. · Zbl 1408.37033
[25] N. P. Fogg,Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Math. 1794, Springer, Berlin, 2002. · Zbl 1014.11015
[26] T. Giordano, I. F. Putnam, and C. F. Skau,Topological orbit equivalence andC∗crossed products, J. Reine Angew. Math. 469 (1995), 51-111. · Zbl 0834.46053
[27] T. Giordano, I. F. Putnam, and C. F. Skau,Full groups of Cantor minimal systems, Israel J. Math. 111 (1999), 285-320. · Zbl 0942.46040
[28] A. Julien and L. Sadun,Tiling deformations, cohomology, and orbit equivalence of tiling spaces, Ann. Henri Poincaré 19 (2018), 3053-3088. · Zbl 1417.37085
[29] J. Kellendonk and I. F. Putnam,The Ruelle-Sullivan map for actions ofRn, Math. Ann. 334 (2006), 693-711. · Zbl 1098.37004
[30] K. H. Kim and F. W. Roush,On the automorphism groups of subshifts, Pure Math. Appl. Ser. B 1 (1990), 203-230. · Zbl 0734.54027
[31] K. Kodaka,Picard groups of irrational rotationC∗-algebras, J. London Math. Soc. (2) 56 (1997), 179-188. · Zbl 0892.46067
[32] J. Krasinkiewicz,Mappings onto circle-like continua, Fund. Math. 91 (1976), 39-49. · Zbl 0329.54031
[33] J. Kwapisz,Rigidity and mapping class group for abstract tiling spaces, Ergodic Theory Dynam. Systems 31 (2011), 1745-1783. · Zbl 1267.37017
[34] J. Kwapisz,Homotopy and dynamics for homeomorphisms of solenoids and Knaster continua, Fund. Math. 168 (2001), 251-278. · Zbl 0984.37018
[35] J. Kwapisz,Topological friction in aperiodic minimalRm-actions, Fund. Math. 207 (2010), 175-178. · Zbl 1190.37009
[36] H. Matui,Finite order automorphisms and dimension groups of Cantor minimal systems, J. Math. Soc. Japan 54 (2002), 135-160. · Zbl 1029.37003
[37] R. McCutcheon,The Gottschalk-Hedlund theorem, Amer. Math. Monthly 106 (1999), 670-672. · Zbl 0987.37010
[38] M. Morse and G. A. Hedlund,Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1-42. · Zbl 0022.34003
[39] W. Narkiewicz,Elementary and Analytic Theory of Algebraic Numbers, 3rd ed., Springer Monogr. Math., Springer, Berlin, 2004. · Zbl 1159.11039
[40] N. Nawata,Fundamental group of simpleC∗-algebras with unique trace III, Canad. J. Math. 64 (2012), 573-587. · Zbl 1252.46050
[41] N. Nawata,Fundamental group of uniquely ergodic Cantor minimal systems, Adv. Math. 230 (2012), 746-758. · Zbl 1285.37003
[42] N. Nawata and Y. Watatani,Fundamental group of simpleC∗-algebras with unique trace, Adv. Math. 225 (2010), 307-318. · Zbl 1202.46068
[43] N. Nawata and Y. Watatani,Fundamental group of simpleC∗-algebras with unique trace II, J. Funct. Anal. 260 (2011), 428-435. · Zbl 1211.46065
[44] I. V. Nikolaev,On embedding of the Bratteli diagram into a surface, in: Recent Advances in Matrix and Operator Theory, Oper. Theory Adv. Appl. 179, Birkhäuser, Basel, 2008, 211-227. · Zbl 1145.46039
[45] J. Olli,Endomorphisms of Sturmian systems and the discrete chair substitution tiling system, Discrete Contin. Dynam. Systems 33 (2013), 4173-4186. · Zbl 1316.37011
[46] B. Parry and D. Sullivan,A topological invariant of flows on1-dimensional spaces, Topology 14 (1975), 297-299. · Zbl 0314.54045
[47] I. F. Putnam,TheC∗-algebras associated with minimal homeomorphisms of the Cantor set, Pacific J. Math. 136 (1989), 329-353. · Zbl 0631.46068
[48] I. Putnam, K. Schmidt, and C. Skau,C∗-algebras associated with Denjoy homeomorphisms of the circle, J. Operator Theory 16 (1986), 99-126. · Zbl 0611.46067
[49] M. Queffélec,Substitution Dynamical Systems—Spectral Analysis, Lecture Notes in Math. 1294, Springer, Berlin, 1987. · Zbl 0642.28013
[50] I. Raeburn and D. P. Williams,Morita Equivalence and Continuous-TraceC∗-algebras, Math. Surveys Monogr. 60, Amer. Math. Soc., Providence, RI, 1998. · Zbl 0922.46050
[51] M. A. Rieffel,C∗-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429. · Zbl 0499.46039
[52] K. Yang,Normal amenable subgroups of the automorphism group of sofic shifts, Ergodic Theory Dynam. Systems (online, 2020).
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.