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The Ellis semigroup of certain constant-length substitutions

Part of: Topological dynamics Semigroups Connections with other structures, applications

Published online by Cambridge University Press:  06 December 2019

PETRA STAYNOVA
PETRA STAYNOVA*
Affiliation:
University of Leeds Faculty of Mathematics and Physical Sciences, Mathematics, Mathematics, Leeds, LS2 9JT, UK email petra.staynova@gmail.com
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Abstract

In this article, we calculate the Ellis semigroup of a certain class of constant-length substitutions. This generalizes a result of Haddad and Johnson [IP cluster points, idempotents, and recurrent sequences. Topology Proc.22 (1997) 213–226] from the binary case to substitutions over arbitrarily large finite alphabets. Moreover, we provide a class of counterexamples to one of the propositions in their paper, which is central to the proof of their main theorem. We give an alternative approach to their result, which centers on the properties of the Ellis semigroup. To do this, we also show a new way to construct an almost automorphic–isometric tower to the maximal equicontinuous factor of these systems, which gives a more particular approach than the one given by Dekking [The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrscheinlichkeitstheor. Verw. Geb.41(3) (1977/78) 221–239].

Keywords

symbolic dynamicstilingstopological dynamics

MSC classification

Primary: 37B10: Symbolic dynamics 54H20: Topological dynamics
Secondary: 20M20: Semigroups of transformations, etc. 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 3 , March 2021 , pp. 935 - 960
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Copyright
© Cambridge University Press, 2019

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References

Allouche, J. P. and Shallit, J.. Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge, 2003.CrossRefGoogle Scholar
Barge, M.. The Ellis semigroup of the Thue–Morse system. Personal communication, 2011.Google Scholar
Blanchard, F., Durand, F. and Maass, A.. Constant-length substitutions and countable scrambled sets. Nonlinearity 17(3) (2004), 817.CrossRefGoogle Scholar
Baake, M. and Grimm, U.. Aperiodic Order (Encyclopedia of Mathematics and its Applications, 149) . Vol. 1. Cambridge University Press, Cambridge, 2013. A mathematical invitation, with a foreword by Roger Penrose.CrossRefGoogle Scholar
Berg, K., Gove, D. and Haddad, K.. Enveloping semigroups and mappings onto the two-shift. Proc. Amer. Math. Soc. 126(3) (1998), 899–905.CrossRefGoogle Scholar
Budak, T., I ̧sik, N., Milnes, P. and Pym, J.. The action of a semisimple Lie group on its maximal compact subgroup. Proc. Amer. Math. Soc. 129(5) (2001), 1525–1534.CrossRefGoogle Scholar
Coven, E. M. and Keane, M. S.. The structure of substitution minimal sets. Trans. Amer. Math. Soc. 162 (1971), 89–102.CrossRefGoogle Scholar
Cobham, A.. Uniform tag sequences. Math. Systems Theory 6 (1972), 164–192.CrossRefGoogle Scholar
Dekking, F. M.. The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrscheinlichkeitstheor. Verw. Geb. 41(3) (1977/78), 221–239.CrossRefGoogle Scholar
de Vries, J.. Elements of Topological Dynamics (Mathematics and its Applications, 257) . Kluwer Academic Publishers Group, Dordrecht, 1993.CrossRefGoogle Scholar
Ellis, D. B. and Ellis, R.. Automorphisms and Equivalence Relations in Topological Dynamics (London Mathematical Society Lecture Note Series, 412) . Cambridge University Press, Cambridge, 2014.CrossRefGoogle Scholar
Ellis, R.. A semigroup associated with a transformation group. Trans. Amer. Math. Soc. 94 (1960), 272–281.CrossRefGoogle Scholar
Pytheas Fogg, N.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794) . Eds. Berthé, V., Ferenczi, S., Mauduit, C. and Siegel, A.. Springer, Berlin, 2002.CrossRefGoogle Scholar
Glasner, E. and Glasner, Y.. A minimal PI cascade with 2c minimal ideals. Ergod. Th. & Dynam. Sys. https://doi.org/10.1017/etds.2018.78, published online 14 September 2018.Google Scholar
Glasner, E.. Short proofs of theorems of Malyutin and Margulis. Proc. Amer. Math. Soc. 145(12) (2017), 5463–5467.CrossRefGoogle Scholar
Glasner, S.. A metric minimal flow whose enveloping semigroup contains finitely many minimal ideals is PI . Israel J. Math. 22(1) (1975), 87–92.CrossRefGoogle Scholar
Glasner, S.. Proximal Flows (Lecture Notes in Mathematics, 517) . Springer, Berlin, 1976.CrossRefGoogle Scholar
Glasner, E.. Minimal nil-transformations of class two. Israel J. Math. 81(1–2) (1993), 31–51.CrossRefGoogle Scholar
Glasner, E. and Megrelishvili, M.. Hereditarily non-sensitive dynamical systems and linear representations. Colloq. Math. 104(2) (2006), 223–283.CrossRefGoogle Scholar
Haddad, K. N.. New limiting notions of the IP type in the enveloping semigroup. Ergod. Th. & Dynam. Sys. 16(4) (1996), 719–733.CrossRefGoogle Scholar
Hedlund, G.. Endomorphisms and automorphisms of the shift dynamical systems. Math. System Theory 3(4) (1969), 320–375.CrossRefGoogle Scholar
Haddad, K. N. and Johnson, A. S. A.. IP cluster points, idempotents, and recurrent sequences. Topology Proc. 22(Spring) (1997), 213–226.Google Scholar
Hindman, N. and Strauss, D.. Algebra in the Stone–Čech Compactification. Vol. 27. de Guyter and Co., Berlin, 1998.CrossRefGoogle Scholar
Keane, M.. Generalized Morse sequences. Z. Wahrscheinlichkeitstheor. Verw. Geb. 10 (1968), 335–353.CrossRefGoogle Scholar
Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
Martin, J. C.. Substitution minimal flows. Amer. J. Math. 93 (1971), 503–526.CrossRefGoogle Scholar
Milnes, P.. Ellis groups and group extensions. Houston J. Math. 12(1) (1986), 87–108.Google Scholar
Milnes, P.. Extension of flows, Ellis groups and groups of Heisenberg type. Math. Nachr. 150 (1991), 97–108.CrossRefGoogle Scholar
Namioka, I.. Ellis groups and compact right topological groups. Conference in Modern Analysis and Probability (New Haven, CT, 1982) (Contemporary Mathematics, 26) . American Mathematical Society, Providence, RI, 1984, pp. 295–300.CrossRefGoogle Scholar
Numakura, K.. On bicompact semigroups. Math. J. Okayama Univ. 1 (1952), 99–108.Google Scholar
Veech, W. A.. Point-distal flows. Amer. J. Math. 92 (1970), 205–242.CrossRefGoogle Scholar
Weisstein, E. W.. Thue–Morse Sequence. MathWorld—A Wolfram Web Resource, http://mathworld.wolfram.com/Thue-MorseSequence.html. Accessed January 2018.Google Scholar