The homology core of matchbox manifolds and invariant measures
HTML articles powered by AMS MathViewer
- by Alex Clark and John Hunton PDF
- Trans. Amer. Math. Soc. 371 (2019), 1771-1793 Request permission
Abstract:
We consider the topology and dynamics associated with a wide class of matchbox manifolds, including spaces of aperiodic tilings and suspensions of higher rank (potentially nonabelian) group actions on zero-dimensional spaces. For such a space we introduce a topological invariant, the homology core, built using an expansion of it as an inverse sequence of simplicial complexes. The invariant takes the form of a monoid equipped with a representation, which in many cases can be used to obtain a finer classification than is possible with the previously developed invariants. When the space is obtained by suspending a topologically transitive action of the fundamental group $\Gamma$ of a closed orientable manifold on a zero-dimensional compact space $Z$, this invariant corresponds to the space of finite Borel measures on $Z$ which are invariant under the action of $\Gamma$. This leads to connections between the rank of the core and the number of invariant, ergodic Borel probability measures for such actions. We illustrate with several examples how these invariants can be calculated and used for topological classification and how it leads to an understanding of the invariant measures.References
- José Aliste-Prieto and Daniel Coronel, Tower systems for linearly repetitive Delone sets, Ergodic Theory Dynam. Systems 31 (2011), no. 6, 1595–1618. MR 2851668, DOI 10.1017/S0143385710000507
- Jared E. Anderson and Ian F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Theory Dynam. Systems 18 (1998), no. 3, 509–537. MR 1631708, DOI 10.1017/S0143385798100457
- Marcy Barge and Beverly Diamond, Homeomorphisms of inverse limit spaces of one-dimensional maps, Fund. Math. 146 (1995), no. 2, 171–187. MR 1314982, DOI 10.4064/fm-146-2-171-187
- Marcy Barge and Beverly Diamond, A complete invariant for the topology of one-dimensional substitution tiling spaces, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1333–1358. MR 1855835, DOI 10.1017/S0143385701001638
- Marcy Barge and Megan Smith, Augmented dimension groups and ordered cohomology, Ergodic Theory Dynam. Systems 29 (2009), no. 1, 1–35. MR 2470625, DOI 10.1017/S0143385708080449
- Marcy Barge and R. F. Williams, Classification of Denjoy continua, Topology Appl. 106 (2000), no. 1, 77–89. MR 1769334, DOI 10.1016/S0166-8641(99)00070-X
- Jean Bellissard, Riccardo Benedetti, and Jean-Marc Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling, Comm. Math. Phys. 261 (2006), no. 1, 1–41. MR 2193205, DOI 10.1007/s00220-005-1445-z
- V. Berthé, W. Steiner and J. Thuswaldner Geometry, dynamics, and arithmetic of $S$-adic shifts, arXiv:1410.0331, 2014.
- S. Bezuglyi, J. Kwiatkowski, K. Medynets, and B. Solomyak, Finite rank Bratteli diagrams: structure of invariant measures, Trans. Amer. Math. Soc. 365 (2013), no. 5, 2637–2679. MR 3020111, DOI 10.1090/S0002-9947-2012-05744-8
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
- Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 1732868, DOI 10.1090/gsm/023
- Alex Clark and John Hunton, Tiling spaces, codimension one attractors and shape, New York J. Math. 18 (2012), 765–796. MR 2991423
- Alex Clark and Steven Hurder, Homogeneous matchbox manifolds, Trans. Amer. Math. Soc. 365 (2013), no. 6, 3151–3191. MR 3034462, DOI 10.1090/S0002-9947-2012-05753-9
- Alex Clark, Steven Hurder, and Olga Lukina, Shape of matchbox manifolds, Indag. Math. (N.S.) 25 (2014), no. 4, 669–712. MR 3217031, DOI 10.1016/j.indag.2014.04.006
- Alex Clark and Lorenzo Sadun, When size matters: subshifts and their related tiling spaces, Ergodic Theory Dynam. Systems 23 (2003), no. 4, 1043–1057. MR 1997967, DOI 10.1017/S0143385702001633
- María Isabel Cortez and Samuel Petite, $G$-odometers and their almost one-to-one extensions, J. Lond. Math. Soc. (2) 78 (2008), no. 1, 1–20. MR 2427048, DOI 10.1112/jlms/jdn002
- F. T. Farrell and L. E. Jones, New attractors in hyperbolic dynamics, J. Differential Geometry 15 (1980), no. 1, 107–133 (1981). MR 602444
- Robbert Johan Fokkink, The structure of trajectories, ProQuest LLC, Ann Arbor, MI, 1991. Thesis (Ph.D.)–Technische Universiteit Delft (The Netherlands). MR 2714473
- Natalie Priebe Frank and Lorenzo Sadun, Fusion: a general framework for hierarchical tilings of $\Bbb {R}^d$, Geom. Dedicata 171 (2014), 149–186. MR 3226791, DOI 10.1007/s10711-013-9893-7
- Harry Furstenberg, Stationary processes and prediction theory, Annals of Mathematics Studies, No. 44, Princeton University Press, Princeton, N.J., 1960. MR 0140151
- D. M. Hardcastle and K. Khanin, On almost everywhere strong convergence of multi-dimensional continued fraction algorithms, Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1711–1733. MR 1804954, DOI 10.1017/S014338570000095X
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
- John Hunton, Spaces of projection method patterns and their cohomology, Mathematics of aperiodic order, Progr. Math., vol. 309, Birkhäuser/Springer, Basel, 2015, pp. 105–135. MR 3381480, DOI 10.1007/978-3-0348-0903-0_{4}
- William B. Jones and Wolfgang J. Thron, Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. Analytic theory and applications; With a foreword by Felix E. Browder; With an introduction by Peter Henrici. MR 595864
- Johannes Kellendonk, The local structure of tilings and their integer group of coinvariants, Comm. Math. Phys. 187 (1997), no. 1, 115–157. MR 1463824, DOI 10.1007/s002200050131
- G. Maloney and D. Rust, Beyond primitivity for one-dimensional substitution subshifts and tiling spaces, Erg. Th. and Dyn. Syst. 38 (2016), no. 3, 1086–1117, doi:10.1017/etds.2016.58.
- Calvin C. Moore and Claude L. Schochet, Global analysis on foliated spaces, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 9, Cambridge University Press, New York, 2006. MR 2202625
- James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
- Samuel Petite, On invariant measures of finite affine type tilings, Ergodic Theory Dynam. Systems 26 (2006), no. 4, 1159–1176. MR 2247636, DOI 10.1017/S0143385706000137
- Nicholas Ormes, Charles Radin, and Lorenzo Sadun, A homeomorphism invariant for substitution tiling spaces, Geom. Dedicata 90 (2002), 153–182. MR 1898159, DOI 10.1023/A:1014942402919
- J. W. Rogers Jr., Inducing approximations homotopic to maps between inverse limits, Fund. Math. 78 (1973), no. 3, 281–289. MR 346728, DOI 10.4064/fm-78-3-250-289
- Boris Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 695–738. MR 1452190, DOI 10.1017/S0143385797084988
- Lorenzo Sadun, Topology of tiling spaces, University Lecture Series, vol. 46, American Mathematical Society, Providence, RI, 2008. MR 2446623, DOI 10.1090/ulect/046
- Lorenzo Sadun and R. F. Williams, Tiling spaces are Cantor set fiber bundles, Ergodic Theory Dynam. Systems 23 (2003), no. 1, 307–316. MR 1971208, DOI 10.1017/S0143385702000949
- Dennis Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225–255. MR 433464, DOI 10.1007/BF01390011
- Richard Swanson and Hans Volkmer, Invariants of weak equivalence in primitive matrices, Ergodic Theory Dynam. Systems 20 (2000), no. 2, 611–626. MR 1756989, DOI 10.1017/S0143385700000316
Additional Information
- Alex Clark
- Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom
- Address at time of publication: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS
- MR Author ID: 639201
- Email: alex.clark@qmul.ac.uk
- John Hunton
- Affiliation: Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, United Kingdom
- Email: john.hunton@durham.ac.uk
- Received by editor(s): July 1, 2016
- Received by editor(s) in revised form: June 28, 2017
- Published electronically: October 11, 2018
- Additional Notes: This work was supported by Leverhulme Trust for International Network grant IN-2013-045.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 1771-1793
- MSC (2010): Primary 37C85; Secondary 28D15, 52C23, 37C70
- DOI: https://doi.org/10.1090/tran/7398
- MathSciNet review: 3894034