Hostname: page-component-7479d7b7d-c9gpj Total loading time: 0 Render date: 2024-07-11T01:10:18.572Z Has data issue: false hasContentIssue false
  • English
  • Français

Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

Published online by Cambridge University Press:  20 September 2016

GREGORY R. MALONEY  and
DAN RUST
GREGORY R. MALONEY
Affiliation:
Newcastle University, Newcastle upon Tyne, NE1 7RU, UK email Gregory.Maloney@newcastle.ac.uk
DAN RUST
Affiliation:
Bielefeld University, Bielefeld 33501, Germany email drust@math.uni-bielefeld.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. We find a characterization of tameness, and use this to prove a slightly stronger version of a result of Durand, which says that the subshift of a minimal substitution is topologically conjugate to the subshift of a primitive substitution. We then extend to the non-minimal setting a result obtained by Anderson and Putnam for primitive substitutions, which says that a substitution tiling space is homeomorphic to an inverse limit of a certain finite graph under a self-map induced by the substitution. We use this result to explore the structure of the lattice of closed invariant subspaces and quotients of a substitution tiling space, for which we compute cohomological invariants that are stronger than the Čech cohomology of the tiling space alone.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 3 , May 2018 , pp. 1086 - 1117
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated C ∗ -algebras. Ergod. Th. & Dynam. Sys. 18(3) (1998), 509–537.Google Scholar
Barge, M. and Diamond, B.. Proximality in Pisot tiling spaces. Fund. Math. 194(3) (2007), 191–238.Google Scholar
Barge, M. and Diamond, B.. Cohomology in one-dimensional substitution tiling spaces. Proc. Amer. Math. Soc. 136(6) (2008), 2183–2191.Google Scholar
Barge, M., Diamond, B. and Holton, C.. Asymptotic orbits of primitive substitutions. Theoret. Comput. Sci. 301(1–3) (2003), 439–450.Google Scholar
Barge, M., Diamond, B., Hunton, J. and Sadun, L.. Cohomology of substitution tiling spaces. Ergod. Th. & Dynam. Sys. 30(6) (2010), 1607–1627.Google Scholar
Bezuglyi, S., Kwiatkowski, J. and Medynets, K.. Aperiodic substitution systems and their Bratteli diagrams. Ergod. Th. & Dynam. Sys. 29(1) (2009), 37–72.Google Scholar
Clark, A. and Hunton, J.. Tiling spaces, codimension one attractors and shape. New York J. Math. 18 (2012), 765–796.Google Scholar
Cortez, M. I. and Solomyak, B.. Invariant measures for non-primitive tiling substitutions. J. Anal. Math. 115 (2011), 293–342.Google Scholar
Damanik, D. and Lenz, D.. Substitution dynamical systems: characterization of linear repetitivity and applications. J. Math. Anal. Appl. 321(2) (2006), 766–780.Google Scholar
Durand, F.. A characterization of substitutive sequences using return words. Discrete Math. 179(1–3) (1998), 89–101.CrossRefGoogle Scholar
Durand, F.. Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. & Dynam. Sys. 20 (2000), 1061–1078.Google Scholar
Gähler, F. and Maloney, G. R.. Cohomology of one-dimensional mixed substitution tiling spaces. Topology Appl. 160(5) (2013), 703–719.Google Scholar
Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
Mossé, B.. Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci. 99(2) (1992), 327–334.Google Scholar
Rust, D.. An uncountable set of tiling spaces with distinct cohomology. Topology Appl. 205 (2016), 58–81.Google Scholar
Sadun, L.. Topology of Tiling Spaces (University Lecture Series, 46) . American Mathematical Society, Providence, RI, 2008.Google Scholar