Interplay between finite topological rank minimal Cantor systems, $\mathcal S$-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
MathSciNet review:
4237953
Full-text PDF
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 ${\mathcal S}$-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 ${\mathcal S}$-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 ${\mathcal S}$-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.
- 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. MR 956049
- 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
- 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. MR 2643706, DOI https://doi.org/10.1017/S0143385709000236
- Mike Boyle and David Handelman, Entropy versus orbit equivalence for minimal homeomorphisms, Pacific J. Math. 164 (1994), no. 1, 1–13. MR 1267499
- S. Bezuglyi, J. Kwiatkowski, and K. Medynets, Aperiodic substitution systems and their Bratteli diagrams, Ergodic Theory Dynam. Systems 29 (2009), no. 1, 37–72. MR 2470626, DOI https://doi.org/10.1017/S0143385708000230
- 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. MR 3020111, DOI https://doi.org/10.1090/S0002-9947-2012-05744-8
- Michael Boshernitzan, A unique ergodicity of minimal symbolic flows with linear block growth, J. Analyse Math. 44 (1984/85), 77–96. MR 801288, DOI https://doi.org/10.1007/BF02790191
- 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. MR 4015135, DOI https://doi.org/10.1017/etds.2017.144
- 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. MR 1967706, DOI https://doi.org/10.1112/S0024610703004320
- María Isabel Cortez, Fabien Durand, and Samuel Petite, Eigenvalues and strong orbit equivalence, Ergodic Theory Dynam. Systems 36 (2016), no. 8, 2419–2440. MR 3570018, DOI https://doi.org/10.1017/etds.2015.26
- 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. MR 3324942, DOI https://doi.org/10.1017/fms.2015.3
- 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. MR 3896204, DOI https://doi.org/10.4171/JEMS/838
- 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. MR 3436754, DOI https://doi.org/10.1017/etds.2015.70
- Sebastian Donoso, Fabien Durand, Alejandro Maass, and Samuel Petite, On automorphism groups of Toeplitz subshifts, Discrete Anal. , posted on (2017), Paper No. 11, 19. MR 3663120, DOI https://doi.org/10.19086/da.1832
- 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. MR 3456604, DOI https://doi.org/10.1017/etds.2014.45
- Fabien Durand, Alexander Frank, and Alejandro Maass, Eigenvalues of minimal Cantor systems, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 3, 727–775. MR 3908764, DOI https://doi.org/10.4171/JEMS/849
- 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. MR 1709427, DOI https://doi.org/10.1017/S0143385799133947
- 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). MR 2996779, DOI https://doi.org/10.1016/j.crma.2012.10.015
- Tomasz Downarowicz and Alejandro Maass, Finite-rank Bratteli-Vershik diagrams are expansive, Ergodic Theory Dynam. Systems 28 (2008), no. 3, 739–747. MR 2422014, DOI https://doi.org/10.1017/S0143385707000673
- Fabien Durand, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998), no. 1-3, 89–101. MR 1489074, DOI https://doi.org/10.1016/S0012-365X%2897%2900029-0
- Fabien Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam. Systems 20 (2000), no. 4, 1061–1078. MR 1779393, DOI https://doi.org/10.1017/S0143385700000584
- 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. MR 2759109
- Bastián Espinoza and Alejandro Maass, On the automorphism group of minimal s-adic subshifts of finite alphabet rank, 2020.
- Sébastien Ferenczi, Rank and symbolic complexity, Ergodic Theory Dynam. Systems 16 (1996), no. 4, 663–682. MR 1406427, DOI https://doi.org/10.1017/S0143385700009032
- 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. MR 2759110
- 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. MR 3830245, DOI https://doi.org/10.1007/s00209-017-1994-9
- 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. MR 1804953, DOI https://doi.org/10.1017/S0143385700000948
- Thierry Giordano, Ian F. Putnam, and Christian F. Skau, Topological orbit equivalence and $C^*$-crossed products, J. Reine Angew. Math. 469 (1995), 51–111. MR 1363826
- Siri-Malén Høynes, Finite-rank Bratteli-Vershik diagrams are expansive—a new proof, Math. Scand. 120 (2017), no. 2, 195–210. MR 3657412, DOI https://doi.org/10.7146/math.scand.a-25613
- 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. MR 1194074, DOI https://doi.org/10.1142/S0129167X92000382
- Antoine Julien, Complexity and cohomology for cut-and-projection tilings, Ergodic Theory Dynam. Systems 30 (2010), no. 2, 489–523. MR 2599890, DOI https://doi.org/10.1017/S0143385709000194
- 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). MR 1872444, DOI https://doi.org/10.1016/S0304-3975%2800%2900435-7
- M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 90, Cambridge University Press, Cambridge, 2002.
- Marston Morse and Gustav A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1–42. MR 745, DOI https://doi.org/10.2307/2371431
- Brigitte Mossé, Puissances de mots et reconnaissabilité des points fixes d’une substitution, Theoret. Comput. Sci. 99 (1992), no. 2, 327–334 (French). MR 1168468, DOI https://doi.org/10.1016/0304-3975%2892%2990357-L
- Donald S. Ornstein, Daniel J. Rudolph, and Benjamin Weiss, Equivalence of measure preserving transformations, Mem. Amer. Math. Soc. 37 (1982), no. 262, xii+116. MR 653094, DOI https://doi.org/10.1090/memo/0262
- Anthony Quas and Luca Zamboni, Periodicity and local complexity, Theoret. Comput. Sci. 319 (2004), no. 1-3, 229–240. MR 2074955, DOI https://doi.org/10.1016/j.tcs.2004.02.026
- Takashi Shimomura, Finite-rank Bratteli-Vershik homeomorphisms are expansive, Proc. Amer. Math. Soc. 145 (2017), no. 10, 4353–4362. MR 3690619, DOI 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
MR Author ID:
628466
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
MR Author ID:
315077
ORCID:
0000-0002-7038-4527
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
MR Author ID:
784469
Email:
samuel.petite@u-picardie.fr
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