Interplay between finite topological rank minimal Cantor systems, -adic subshifts and their complexity
Authors:
Sebastián Donoso, Fabien Durand, Alejandro Maass and Samuel Petite
Journal:
Trans. Amer. Math. Soc. 374 (2021), 3453-3489
MSC (2020):
Primary 37B10, 68R15
DOI:
https://doi.org/10.1090/tran/8315
Published electronically:
February 23, 2021
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Abstract | References | Similar Articles | Additional Information
Abstract: Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable -adic subshifts. This is done by establishing necessary and sufficient conditions for a minimal subshift to be of finite topological rank. As an application, we show that minimal subshifts with non-superlinear complexity (like many classical zero-entropy examples) have finite topological rank. Conversely, we analyze the complexity of
-adic subshifts and provide sufficient conditions for a finite topological rank subshift to have a non-superlinear complexity. This includes minimal Cantor systems given by Bratteli-Vershik representations whose tower levels have proportional heights and the so-called left to right
-adic subshifts. We also show that finite topological rank does not imply non-superlinear complexity. In the particular case of topological rank two subshifts, we prove their complexity is always subquadratic along a subsequence and their automorphism group is trivial.
- [Aus88] Joseph Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. {956049}
- [BCBD+] V. Berthé, P. Cecchi-Bernales, F. Durand, J. Leroy, D. Perrin, and S. Petite, On the dimension group of unimodular S-adic subshifts, Monatsh. Math. (2021), DOI 10.1007/s00605-020-01488-3
- [BDM10] Xavier Bressaud, Fabien Durand, and Alejandro Maass, On the eigenvalues of finite rank Bratteli-Vershik dynamical systems, Ergodic Theory Dynam. Systems 30 (2010), no. 3, 639–664. {2643706}, https://doi.org/10.1017/S0143385709000236
- [BH94] Mike Boyle and David Handelman, Entropy versus orbit equivalence for minimal homeomorphisms, Pacific J. Math. 164 (1994), no. 1, 1–13. {1267499}
- [BKM08] S. Bezuglyi, J. Kwiatkowski, and K. Medynets, Aperiodic substitution systems and their Bratteli diagrams, Ergodic Theory Dynam. Systems 29 (2009), no. 1, 37–72. {2470626}, https://doi.org/10.1017/S0143385708000230
- [BKMS13] S. Bezuglyi, J. Kwiatkowski, K. Medynets, and B. Solomyak, Finite rank Bratteli diagrams: structure of invariant measures, Trans. Amer. Math. Soc. 365 (2013), no. 5, 2637–2679. {3020111}, https://doi.org/10.1090/S0002-9947-2012-05744-8
- [Bos85] Michael Boshernitzan, A unique ergodicity of minimal symbolic flows with linear block growth, J. Analyse Math. 44 (1984/85), 77–96. {801288}, https://doi.org/10.1007/BF02790191
- [BSTY19] Valérie Berthé, Wolfgang Steiner, Jörg M. Thuswaldner, and Reem Yassawi, Recognizability for sequences of morphisms, Ergodic Theory Dynam. Systems 39 (2019), no. 11, 2896–2931. {4015135}, https://doi.org/10.1017/etds.2017.144
- [CDHM03] Maria Isabel Cortez, Fabien Durand, Bernard Host, and Alejandro Maass, Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems, J. London Math. Soc. (2) 67 (2003), no. 3, 790–804. {1967706}, https://doi.org/10.1112/S0024610703004320
- [CDP16] María Isabel Cortez, Fabien Durand, and Samuel Petite, Eigenvalues and strong orbit equivalence, Ergodic Theory Dynam. Systems 36 (2016), no. 8, 2419–2440. {3570018}, https://doi.org/10.1017/etds.2015.26
- [CK15] Van Cyr and Bryna Kra, The automorphism group of a shift of linear growth: beyond transitivity, Forum Math. Sigma 3 (2015), Paper No. e5, 27. {3324942}, https://doi.org/10.1017/fms.2015.3
- [CK19] Van Cyr and Bryna Kra, Counting generic measures for a subshift of linear growth, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 2, 355–380. {3896204}, https://doi.org/10.4171/JEMS/838
- [DDMP16] Sebastián Donoso, Fabien Durand, Alejandro Maass, and Samuel Petite, On automorphism groups of low complexity subshifts, Ergodic Theory Dynam. Systems 36 (2016), no. 1, 64–95. {3436754}, https://doi.org/10.1017/etds.2015.70
- [DDMP17] Sebastian Donoso, Fabien Durand, Alejandro Maass, and Samuel Petite, On automorphism groups of Toeplitz subshifts, Discrete Anal. , posted on (2017), Paper No. 11, 19. {3663120}, https://doi.org/10.19086/da.1832
- [DFM15] Fabien Durand, Alexander Frank, and Alejandro Maass, Eigenvalues of Toeplitz minimal systems of finite topological rank, Ergodic Theory Dynam. Systems 35 (2015), no. 8, 2499–2528. {3456604}, https://doi.org/10.1017/etds.2014.45
- [DFM19] Fabien Durand, Alexander Frank, and Alejandro Maass, Eigenvalues of minimal Cantor systems, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 3, 727–775. {3908764}, https://doi.org/10.4171/JEMS/849
- [DHS99] F. Durand, B. Host, and C. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam. Systems 19 (1999), no. 4, 953–993. {1709427}, https://doi.org/10.1017/S0143385799133947
- [DL12] Fabien Durand and Julien Leroy, 𝑆-adic conjecture and Bratteli diagrams, C. R. Math. Acad. Sci. Paris 350 (2012), no. 21-22, 979–983 (English, with English and French summaries). {2996779}, https://doi.org/10.1016/j.crma.2012.10.015
- [DM08] Tomasz Downarowicz and Alejandro Maass, Finite-rank Bratteli-Vershik diagrams are expansive, Ergodic Theory Dynam. Systems 28 (2008), no. 3, 739–747. {2422014}, https://doi.org/10.1017/S0143385707000673
- [Dur98] Fabien Durand, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998), no. 1-3, 89–101. {1489074}, https://doi.org/10.1016/S0012-365X(97)00029-0
- [Dur00] Fabien Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam. Systems 20 (2000), no. 4, 1061–1078. {1779393}, https://doi.org/10.1017/S0143385700000584
- [Dur10] Fabien Durand, Combinatorics on Bratteli diagrams and dynamical systems, Combinatorics, automata and number theory, Encyclopedia Math. Appl., vol. 135, Cambridge Univ. Press, Cambridge, 2010, pp. 324–372. {2759109}
- [EM20] Bastián Espinoza and Alejandro Maass, On the automorphism group of minimal s-adic subshifts of finite alphabet rank, 2020.
- [Fer96] Sébastien Ferenczi, Rank and symbolic complexity, Ergodic Theory Dynam. Systems 16 (1996), no. 4, 663–682. {1406427}, https://doi.org/10.1017/S0143385700009032
- [FM10] Sébastien Ferenczi and Thierry Monteil, Infinite words with uniform frequencies, and invariant measures, Combinatorics, automata and number theory, Encyclopedia Math. Appl., vol. 135, Cambridge Univ. Press, Cambridge, 2010, pp. 373–409. {2759110}
- [GHH18] T. Giordano, D. Handelman, and M. Hosseini, Orbit equivalence of Cantor minimal systems and their continuous spectra, Math. Z. 289 (2018), no. 3-4, 1199–1218. {3830245}, https://doi.org/10.1007/s00209-017-1994-9
- [GJ00] Richard Gjerde and Ørjan Johansen, Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows, Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1687–1710. {1804953}, https://doi.org/10.1017/S0143385700000948
- [GPS95] Thierry Giordano, Ian F. Putnam, and Christian F. Skau, Topological orbit equivalence and 𝐶*-crossed products, J. Reine Angew. Math. 469 (1995), 51–111. {1363826}
- [Høy17] Siri-Malén Høynes, Finite-rank Bratteli-Vershik diagrams are expansive—a new proof, Math. Scand. 120 (2017), no. 2, 195–210. {3657412}, https://doi.org/10.7146/math.scand.a-25613
- [HPS92] Richard H. Herman, Ian F. Putnam, and Christian F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827–864. {1194074}, https://doi.org/10.1142/S0129167X92000382
- [Jul10] Antoine Julien, Complexity and cohomology for cut-and-projection tilings, Ergodic Theory Dynam. Systems 30 (2010), no. 2, 489–523. {2599890}, https://doi.org/10.1017/S0143385709000194
- [KM02] Juhani Karhumäki and Ján Maňuch, Multiple factorizations of words and defect effect, Theoret. Comput. Sci. 273 (2002), no. 1-2, 81–97. WORDS (Rouen, 1999). {1872444}, https://doi.org/10.1016/S0304-3975(00)00435-7
- [Lot02] M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 90, Cambridge University Press, Cambridge, 2002.
- [MH40] Marston Morse and Gustav A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1–42. {745}, https://doi.org/10.2307/2371431
- [Mos92] Brigitte Mossé, Puissances de mots et reconnaissabilité des points fixes d’une substitution, Theoret. Comput. Sci. 99 (1992), no. 2, 327–334 (French). {1168468}, https://doi.org/10.1016/0304-3975(92)90357-L
- [ORW82] Donald S. Ornstein, Daniel J. Rudolph, and Benjamin Weiss, Equivalence of measure preserving transformations, Mem. Amer. Math. Soc. 37 (1982), no. 262, xii+116. {653094}, https://doi.org/10.1090/memo/0262
- [QZ04] Anthony Quas and Luca Zamboni, Periodicity and local complexity, Theoret. Comput. Sci. 319 (2004), no. 1-3, 229–240. {2074955}, https://doi.org/10.1016/j.tcs.2004.02.026
- [Shi17] Takashi Shimomura, Finite-rank Bratteli-Vershik homeomorphisms are expansive, Proc. Amer. Math. Soc. 145 (2017), no. 10, 4353–4362. {3690619}, https://doi.org/10.1090/proc/13575
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Additional Information
Sebastián Donoso
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile & UMI-CNRS 2807, Beauchef 851, Santiago, Chile
Email:
sdonoso@dim.uchile.cl
Fabien Durand
Affiliation:
Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens cedex 1, France
Email:
fabien.durand@u-picardie.fr
Alejandro Maass
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile & UMI-CNRS 2807, Beauchef 851, Santiago, Chile
Email:
amaass@dim.uchile.cl
Samuel Petite
Affiliation:
Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens cedex 1, France
Email:
samuel.petite@u-picardie.fr
DOI:
https://doi.org/10.1090/tran/8315
Keywords:
${\mathcal{S}}$-adic subshifts,
minimal Cantor systems,
finite topological rank,
recognizability,
complexity
Received by editor(s):
March 13, 2020
Received by editor(s) in revised form:
August 31, 2020
Published electronically:
February 23, 2021
Additional Notes:
This research was partially supported by grants PIA-ANID AFB 170001 & Fondap 15090007, and grant ANID/MEC/80180045 hosted by University of O’Higgins.
The first and the third authors thank the hospitality of the LAMFA UMR 7352 CNRS-UPJV and the “poste rouge” CNRS program.
Article copyright:
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American Mathematical Society