Hostname: page-component-cb9f654ff-fg9bn Total loading time: 0 Render date: 2025-08-23T18:00:15.067Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamical properties of minimal Ferenczi subshifts

Part of: Topological dynamics Ergodic theory

Published online by Cambridge University Press:  22 February 2023

FELIPE ARBULÚ  and
FABIEN DURAND Open the ORCID record for FABIEN DURAND [Opens in a new window]
FELIPE ARBULÚ
Affiliation:
Laboratoire Amiénois de Mathématique Fondamentale et Apliquée, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens cedex 1, France (e-mail: felipe.arbulu@u-picardie.fr)
FABIEN DURAND*
Affiliation:
Laboratoire Amiénois de Mathématique Fondamentale et Apliquée, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens cedex 1, France (e-mail: felipe.arbulu@u-picardie.fr)
*
e-mail: fabien.durand@u-picardie.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide an explicit $\mathcal {S}$-adic representation of rank-one subshifts with bounded spacers and call the subshifts obtained in this way ‘minimal Ferenczi subshifts’. We aim to show that this approach is very convenient to study the dynamical behavior of rank-one systems. For instance, we compute their topological rank, the strong and the weak orbit equivalence class. We observe that they have an induced system that is a Toeplitz subshift having discrete spectrum. We also characterize continuous and non-continuous eigenvalues of minimal Ferenczi subshifts.

Keywords

Ferenczi subshiftsrank-one subshiftsS-adic subshiftsminimal Cantor systems

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 37A05: Measure-preserving transformations

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 12 , December 2023 , pp. 3923 - 3970
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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.)

Article purchase

Temporarily unavailable

References

Avila, A. and Delecroix, V.. Weak mixing directions in non-arithmetic Veech surfaces. J. Amer. Math. Soc. 29(4) (2016), 1167–1208.10.1090/jams/856CrossRefGoogle Scholar
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
Avila, A. and Forni, G.. Weak mixing for interval exchange transformations and translation flows. Ann. of Math. (2) 165(2) (2007), 637–664.CrossRefGoogle Scholar
Adams, T., Ferenczi, S. and Petersen, K.. Constructive symbolic presentations of rank one measure-preserving systems. Colloq. Math. 150(2) (2017), 243–255.CrossRefGoogle Scholar
Auslander, J.. Minimal Flows and their Extensions [(North-Holland Mathematics Studies, 153), Notas de Matemática [Mathematical Notes], 122]. North-Holland Publishing Co., Amsterdam, 1988.Google Scholar
Berthé, V., Cecchi Bernales, P., Durand, F., Leroy, J., Perrin, D. and Petite, S.. On the dimension group of unimodular S-adic subshifts. Monatsh. Math. 194 (2021), 687–717.CrossRefGoogle Scholar
Berthé, V. and Delecroix, V.. Beyond substitutive dynamical systems: S-adic expansions. Numeration and Substitution 2012 (RIMS Kôkyûroku Bessatsu, B46). Eds. Akiyama, S., Berthé, V., Rao, H. and Komatsu, T.. Research Institute for Mathematical Sciences (RIMS), Kyoto, 2014, pp. 81–123.Google Scholar
Bressaud, X., Durand, F. and Maass, A.. Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems. J. Lond. Math. Soc. (2) 72(3) (2005), 799–816.CrossRefGoogle Scholar
Bressaud, X., Durand, F. and Maass, A.. On the eigenvalues of finite rank Bratteli–Vershik dynamical systems. Ergod. Th. & Dynam. Sys. 30(3) (2010), 639–664.CrossRefGoogle Scholar
Bezuglyi, S., Kwiatkowski, J., Medynets, K. and Solomyak, B.. Finite rank Bratteli diagrams: structure of invariant measures. Trans. Amer. Math. Soc. 365(5) (2013), 2637–2679.CrossRefGoogle Scholar
Bourbaki, N.. Éléments de mathématique. Première partie. Fascicule VI. Livre II: Algèbre. Chapitre 2: Algèbre linéaire (Entièrement Refondue, Actualités Sci. Indust., 1236), Troisième édition. Hermann, Paris, 1962.Google Scholar
Bourgain, J.. On the spectral type of Ornstein’s class one transformations. Israel J. Math. 84(1–2) (1993), 53–63.CrossRefGoogle Scholar
Berthé, V., Steiner, W., Thuswaldner, J. M. and Yassawi, R.. Recognizability for sequences of morphisms. Ergod. Th. & Dynam. Sys. 39(11) (2019), 2896–2931 CrossRefGoogle Scholar
Chacon, R. V.. A geometric construction of measure preserving transformations. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, CA, 1965/66), Volume II: Contributions to Probability Theory, Part 2. Eds. Le Cam, L. M. and Neyman, J.. University of California Press, Berkeley, CA, 1967, pp. 335–360.Google Scholar
Creutz, D., Pavlov, R. and Rodock, S.. Measure-theoretically mixing subshifts with low complexity. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2022.42. Published online 13 June 2022.CrossRefGoogle Scholar
Creutz, D.. Measure-theoretically mixing subshifts of minimal word complexity. Preprint, 2022, arXiv:2206.10047.CrossRefGoogle 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.CrossRefGoogle Scholar
Donoso, S., Durand, F., Maass, A. and Petite, S.. Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity. Trans. Amer. Math. Soc. 374(5) (2021), 3453–3489.CrossRefGoogle Scholar
Dekking, F. M.. The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41(3) (1978), 221–239.CrossRefGoogle Scholar
Durand, F., Frank, A. and Maass, A.. Eigenvalues of Toeplitz minimal systems of finite topological rank. Ergod. Th. & Dynam. Sys. 35(8) (2015), 2499–2528.CrossRefGoogle Scholar
Durand, F., Frank, A. and Maass, A.. Eigenvalues of minimal Cantor systems. J. Eur. Math. Soc. (JEMS) 21(3) (2019), 727–775.10.4171/JEMS/849CrossRefGoogle Scholar
Downarowicz, T. and Glasner, E.. Isomorphic extensions and applications. Topol. Methods Nonlinear Anal. 48(1) (2016), 321–338.Google Scholar
Durand, F., Host, B. and Skau, C.. Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergod. Th. & Dynam. Sys. 19(4) (1999), 953–993.CrossRefGoogle Scholar
Durand, F. and Leroy, J.. S-adic conjecture and Bratteli diagrams. C. R. Math. Acad. Sci. Paris 350(21–22) (2012), 979–983 10.1016/j.crma.2012.10.015CrossRefGoogle Scholar
Downarowicz, T. and Maass, A.. Finite-rank Bratteli–Vershik diagrams are expansive. Ergod. Th. & Dynam. Sys. 28(3) (2008), 739–747.CrossRefGoogle Scholar
Durand, F. and Perrin, D.. Dimension Groups And Dynamical Systems: Substitutions, Bratteli Diagrams and Cantor Systems (Cambridge Studies in Advanced Mathematics, 196). Cambridge University Press, Cambridge, 2022.CrossRefGoogle Scholar
Durand, F.. Linearly recurrent subshifts have a finite number of nonperiodic subshift factors. Ergod. Th. & Dynam. Sys. 20(4) (2000), 1061–1078.CrossRefGoogle Scholar
Durand, F.. Combinatorics on Bratteli diagrams and dynamical systems. Combinatorics, Automata and Number Theory (Encyclopedia of Mathematics and its Applications, 135). Eds. Berthé, V. and Rigo, M.. Cambridge University Press, Cambridge, 2010, pp. 324–372.CrossRefGoogle Scholar
Espinoza, B.. Symbolic factors of S-adic subshifts of finite alphabet rank. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2022.21. Published online 17 March 2022.CrossRefGoogle Scholar
Ferenczi, S.. Rank and symbolic complexity. Ergod. Th. & Dynam. Sys. 16(4) (1996), 663–682.CrossRefGoogle Scholar
Ferenczi, S.. Systems of finite rank. Colloq. Math. 73(1) (1997), pp. 35–65.10.4064/cm-73-1-35-65CrossRefGoogle Scholar
Gao, S. and Hill, A.. A model for rank one measure preserving transformations. Topology Appl. 174 (2014), 25–40.CrossRefGoogle Scholar
Gao, S. and Hill, A.. Bounded rank-1 transformations. J. Anal. Math. 129 (2016), 341–365.CrossRefGoogle Scholar
Gao, S. and Hill, A.. Topological isomorphism for rank-1 systems. J. Anal. Math. 128 (2016), 1–49.CrossRefGoogle Scholar
Gao, S. and Hill, A.. Disjointness between bounded rank-one transformations. Colloq. Math. 164(1) (2021), 91–121.10.4064/cm8081-2-2020CrossRefGoogle Scholar
Gjerde, R. and Johansen, O.. Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows. Ergod. Th. & Dynam. Sys. 20(6) (2000), 1687–1710.10.1017/S0143385700000948CrossRefGoogle Scholar
Giordano, T., Putnam, I. F. and Skau, C. F.. Topological orbit equivalence and C ${}^{\ast }$ -crossed products. J. Reine Angew. Math. 469 (1995), 51–111.Google Scholar
García-Ramos, F., Jäger, T. and Ye, X.. Mean equicontinuity, almost automorphy and regularity. Israel J. Math. 243(1) (2021), 155–183.CrossRefGoogle Scholar
Gao, S. and Ziegler, C.. Topological mixing properties of rank-one subshifts. Trans. London Math. Soc. 6(1) (2019), 1–21.CrossRefGoogle Scholar
Gao, S. and Ziegler, C.. Topological factors of rank-one subshifts. Proc. Amer. Math. Soc. Ser. B 7 (2020), 118–126.CrossRefGoogle Scholar
Halmos, P. R. and von Neumann, J.. Operator methods in classical mechanics. II. Ann. of Math. (2) 43 (1942), 332–350.10.2307/1968872CrossRefGoogle Scholar
Host, B.. Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable. Ergod. Th. & Dynam. Sys. 6(4) (1986), 529–540.CrossRefGoogle Scholar
Herman, R. H., Putnam, I. F. and Skau, C. F.. Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3(6) (1992), 827–864.10.1142/S0129167X92000382CrossRefGoogle Scholar
del Junco, A.. Transformations with discrete spectrum are stacking transformations. Canad. J. Math. 28(4) (1976), 836–839.CrossRefGoogle Scholar
Kalikow, S. A.. Twofold mixing implies threefold mixing for rank one transformations. Ergod. Th. & Dynam. Sys. 4(2) (1984), 237–259.CrossRefGoogle Scholar
King, J.. The commutant is the weak closure of the powers, for rank-1 transformations. Ergod. Th. & Dynam. Sys. 6(3) (1986), 363–384.CrossRefGoogle Scholar
King, J. L.. Joining-rank and the structure of finite rank mixing transformations. J. Anal. Math. 51 (1988), 182–227.CrossRefGoogle Scholar
Kenyon, R., Sadun, L. and Solomyak, B.. Topological mixing for substitutions on two letters. Ergod. Th. & Dynam. Sys. 25(6) (2005), 1919–1934.CrossRefGoogle Scholar
Leroy, J.. An S-adic characterization of minimal subshifts with first difference of complexity $1\le p\left(n+1\right)-p(n)\le 2$ . Discrete Math. Theor. Comput. Sci. 16(1) (2014), 233–286.Google Scholar
Li, J., Tu, S. and Ye, X.. Mean equicontinuity and mean sensitivity. Ergod. Th. & Dynam. Sys. 35(8) (2015), 2587–2612.CrossRefGoogle Scholar
Nadkarni, M. G.. Spectral Theory of Dynamical Systems (Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]). Birkhäuser Verlag, Basel, 1998.Google Scholar
Ornstein, D. S.. On the root problem in ergodic theory. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (University of California, Berkeley, CA, 1970/1971), Volume II: Probability Theory. Eds. Le Cam, L. M., Neyman, J. and Scott, E. L.. University of California Press, Berkeley, CA, 1972, pp. 347–356.Google Scholar
Ornstein, D. S., Rudolph, D. J. and Weiss, B.. Equivalence of measure preserving transformations. Mem. Amer. Math. Soc. 37(262) (1982), xii+116.Google Scholar
Queffélec, M.. Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics, 1294). Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
Ryzhikov, V. V.. Measure-preserving rank one transformations. Trans. Moscow Math. Soc. 81 (2020), 229–259.CrossRefGoogle Scholar
Veech, W. A.. The metric theory of interval exchange transformations. I. Generic spectral properties. Amer. J. Math. 106(6) (1984), 1331–1359.CrossRefGoogle Scholar