Skip to main content Accessibility help
×
×
Home

Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

  • GREGORY R. MALONEY (a1) and DAN RUST (a2)

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.

Copyright

References

Hide All
[1] Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated C -algebras. Ergod. Th. & Dynam. Sys. 18(3) (1998), 509537.
[2] Barge, M. and Diamond, B.. Proximality in Pisot tiling spaces. Fund. Math. 194(3) (2007), 191238.
[3] Barge, M. and Diamond, B.. Cohomology in one-dimensional substitution tiling spaces. Proc. Amer. Math. Soc. 136(6) (2008), 21832191.
[4] Barge, M., Diamond, B. and Holton, C.. Asymptotic orbits of primitive substitutions. Theoret. Comput. Sci. 301(1–3) (2003), 439450.
[5] Barge, M., Diamond, B., Hunton, J. and Sadun, L.. Cohomology of substitution tiling spaces. Ergod. Th. & Dynam. Sys. 30(6) (2010), 16071627.
[6] Bezuglyi, S., Kwiatkowski, J. and Medynets, K.. Aperiodic substitution systems and their Bratteli diagrams. Ergod. Th. & Dynam. Sys. 29(1) (2009), 3772.
[7] Clark, A. and Hunton, J.. Tiling spaces, codimension one attractors and shape. New York J. Math. 18 (2012), 765796.
[8] Cortez, M. I. and Solomyak, B.. Invariant measures for non-primitive tiling substitutions. J. Anal. Math. 115 (2011), 293342.
[9] Damanik, D. and Lenz, D.. Substitution dynamical systems: characterization of linear repetitivity and applications. J. Math. Anal. Appl. 321(2) (2006), 766780.
[10] Durand, F.. A characterization of substitutive sequences using return words. Discrete Math. 179(1–3) (1998), 89101.
[11] Durand, F.. Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. & Dynam. Sys. 20 (2000), 10611078.
[12] Gähler, F. and Maloney, G. R.. Cohomology of one-dimensional mixed substitution tiling spaces. Topology Appl. 160(5) (2013), 703719.
[13] Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.
[14] Mossé, B.. Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci. 99(2) (1992), 327334.
[15] Rust, D.. An uncountable set of tiling spaces with distinct cohomology. Topology Appl. 205 (2016), 5881.
[16] Sadun, L.. Topology of Tiling Spaces (University Lecture Series, 46) . American Mathematical Society, Providence, RI, 2008.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed