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Word complexity of (measure-theoretically) weakly mixing rank-one subshifts

Part of: Topological dynamics Ergodic theory

Published online by Cambridge University Press:  05 July 2023

DARREN CREUTZ
DARREN CREUTZ*
Affiliation:
Department of Mathematics, US Naval Academy, Annapolis, USA
*
email: creutz@usna.edu
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Abstract

We exhibit, for arbitrary $\epsilon> 0$, subshifts admitting weakly mixing (probability) measures with word complexity p satisfying $\limsup p(q) / q < 1.5 + \epsilon $. For arbitrary $f(q) \to \infty $, said subshifts can be made to satisfy $p(q) < q + f(q)$ infinitely often. We establish that every subshift associated to a rank-one transformation (on a probability space) which is not an odometer satisfies $\limsup p(q) - 1.5q = \infty $ and that this is optimal for rank-ones.

Keywords

symbolic dynamicsword complexityweak mixingrank-one transformations

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 37A25: Ergodicity, mixing, rates of mixing
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 5 , May 2024 , pp. 1330 - 1366
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Adams, T. M.. Smorodinsky’s conjecture on rank-one mixing. Proc. Amer. Math. Soc. 126(3) (1998), 739–744.10.1090/S0002-9939-98-04082-9CrossRefGoogle Scholar
Adams, T., Ferenczi, S. and Petersen, K.. Constructive symbolic presentations of rank one measure-preserving systems. Colloq. Math. 150(2) (2017), 243–255.10.4064/cm7124-3-2017CrossRefGoogle Scholar
Cassaigne, J.. Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. Simon Stevin 4(1) (1997), 67–88.10.36045/bbms/1105730624CrossRefGoogle Scholar
Cassaigne, J.. Sequences with grouped factors. Developments in Language Theory III. Publications of Aristotle University of Thessaloniki, 1998, pp. 211–222.Google Scholar
Cassaigne, J., Frid, A. E., Puzynina, S. and Zamboni, L. Q.. A characterization of words of linear complexity. Proc. Amer. Math. Soc. 147(7) (2019), 3103–3115.10.1090/proc/14440CrossRefGoogle Scholar
Coven, E. M. and Hedlund, G. A.. Sequences with minimal block growth. Math. Syst. Theory 7 (1973), 138–153.10.1007/BF01762232CrossRefGoogle Scholar
Cyr, V. and Kra, B.. Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc. (JEMS) 21(2) (2019), 355–380.10.4171/jems/838CrossRefGoogle Scholar
Cyr, V. and Kra, B.. The automorphism group of a shift of slow growth is amenable. Ergod. Th. & Dynam. Sys. 40(7) (2020), 1788–1804.10.1017/etds.2018.126CrossRefGoogle Scholar
Cyr, V. and Kra, B.. Realizing ergodic properties in zero entropy subshifts. Israel J. Math. 240(1) (2020), 119–148.10.1007/s11856-020-2055-3CrossRefGoogle Scholar
Creutz, D., Pavlov, R. and Rodock, S.. Measure-theoretically mixing subshifts with low complexity. Ergod. Th. & Dynam. Sys. 43 (2022), 2293–2316.10.1017/etds.2022.42CrossRefGoogle Scholar
Creutz, D.. Mixing on stochastic staircase tansformations. Studia Math. 257(2) (2021), 121–153.10.4064/sm8063-8-2020CrossRefGoogle Scholar
Creutz, D.. Measure-theoretically mixing subshifts of minimal word complexity. Preprint, 2023, arXiv:2206.10047.Google Scholar
Creutz, D. and Silva, C. E.. Mixing on rank-one transformations. Studia Math. 199(1) (2010), 43–72.10.4064/sm199-1-4CrossRefGoogle Scholar
Danilenko, A. I.. Actions of finite rank: weak rational ergodicity and partial rigidity. Ergod. Th. & Dynam. Sys. 36(7) (2016), 2138–2171.10.1017/etds.2015.5CrossRefGoogle Scholar
Danilenko, A. I.. Rank-one actions, their $(C,F)$ -models and constructions with bounded parameters. J. Anal. Math. 139(2) (2019), 697–749.10.1007/s11854-023-0075-8CrossRefGoogle Scholar
Donoso, S., Durand, F., Maass, A. and Petite, S.. On automorphism groups of low complexity subshifts. Ergod. Th. & Dynam. Sys. 36(1) (2016), 64–95.10.1017/etds.2015.70CrossRefGoogle Scholar
del Junco, A.. A transformation with simple spectrum which is not rank one. Canad. J. Math. 29(3) (1977), 655–663.10.4153/CJM-1977-067-7CrossRefGoogle Scholar
Dykstra, A., Ormes, N. and Pavlov, R.. Subsystems of transitive subshifts with linear complexity. Ergod. Th. & Dynam. Sys. 42 (2021), 1967–1993.10.1017/etds.2021.8CrossRefGoogle Scholar
Ferenczi, S.. Les transformations de Chacon combinatoire, structure géométrique, lien avec les systèmes de complexité $2n+1$ . Bull. Soc. Math. France 123(2) (1995), 271–292 (in France).10.24033/bsmf.2260CrossRefGoogle Scholar
Ferenczi, S.. Rank and symbolic complexity. Ergod. Th. & Dynam. Sys. 16(4) (1996), 663–682.10.1017/S0143385700009032CrossRefGoogle Scholar
Ferenczi, S.. Systems of finite rank. Colloq. Math. 73(1) (1997), 35–65.10.4064/cm-73-1-35-65CrossRefGoogle Scholar
Foreman, M., Gao, S., Hill, A., Silva, C. E. and Weiss, B.. Rank one transformations, odometers and finite factors. Israel J. Math. doi: https://doi.org/10.1007/s11856-022-2451-y. Published online 27 December 2022.CrossRefGoogle Scholar
Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981; M. B. Porter, Lectures.10.1515/9781400855162CrossRefGoogle Scholar
Gao, S. and Hill, A.. Bounded rank-1 transformations. J. Anal. Math. 129 (2016), 341–365.10.1007/s11854-016-0024-xCrossRefGoogle Scholar
Gao, S. and Hill, A.. Topological isomorphism for rank-1 systems. J. Anal. Math. 128 (2016), 1–49.10.1007/s11854-016-0001-4CrossRefGoogle Scholar
Gao, S. and Ziegler, C.. Topological mixing properties of rank-one subshifts. Trans. London Math. Soc. 6(1) (2019), 1–21.10.1112/tlm3.12016CrossRefGoogle Scholar
Heinis, A.. The P(n)/n-function for bi-infinite words. Theor. Comput. Sci. 273(1–2) (2002), 35–46.10.1016/S0304-3975(00)00432-1CrossRefGoogle Scholar
Kalikow, S. A.. Twofold mixing implies threefold mixing for rank one transformations. Ergod. Th. & Dynam. Sys. 4(2) (1984), 237–259.10.1017/S014338570000242XCrossRefGoogle Scholar
Morse, M. and Hedlund, G. A.. Symbolic dynamics. Amer. J. Math. 60(4) (1938), 815–866.10.2307/2371264CrossRefGoogle Scholar
Morse, M. and Hedlund, G. A.. Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940), 1–42.10.2307/2371431CrossRefGoogle Scholar
Ormes, N. and Pavlov, R.. On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). Eds. J. Hawkins, R. Rosetti and J. Wiseman. American Mathematical Society, Providence, RI, 2019, pp. 125–137.10.1090/conm/736/14833CrossRefGoogle Scholar
Pavlov, R. and Schmeiding, S.. Local finiteness and automorphism groups of low complexity subshifts. Ergod. Th. & Dynam. Sys. 43 (2023), 1980–2001.10.1017/etds.2022.7CrossRefGoogle Scholar
Rote, G.. Sequences with subword complexity $2n$ . J. Number Theory 46(2) (1994), 196–213.10.1006/jnth.1994.1012CrossRefGoogle Scholar
Ryzhikov, V. V.. Bounded ergodic constructions, disjointness, and weak limits of powers. Trans. Moscow Math. Soc. 74 (2013), 165–171.10.1090/S0077-1554-2014-00214-4CrossRefGoogle Scholar
Silva, C. E.. Invitation to Ergodic Theory (Student Mathematical Library, 42). American Mathematical Society, Providence, RI, 2008.Google Scholar