Homomorphisms between multidimensional constant-shape substitutions

automorphism groups; digit tiles; Homomorphisms; nondeterministic directions; substitutive subshifts; Geometry and Topology; Discrete Mathematics and Combinatorics

Abstract :

[en] We study a class of Zd -substitutive subshifts, including a large family of constant-length substitutions, and homomorphisms between them, i.e., factors modulo isomorphisms of Zd . We prove that any measurable factor map and even any homomorphism associated to a matrix commuting with the expansion matrix, induces a continuous one. We also get strong restrictions on the normalizer group, proving that any endomorphism is invertible, the normalizer group is virtually generated by the shift action and the quotient of the normalizer group by the automorphisms is restricted by the digit tile of the substitution.

Disciplines :

Mathematics

Author, co-author :

Cabezas Aros, Christopher ; Université de Liège - ULiège > Mathematics ; Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, Amiens, France

Language :

English

Title :

Homomorphisms between multidimensional constant-shape substitutions

Publication date :

2023

Journal title :

Groups, Geometry, and Dynamics

ISSN :

1661-7207

eISSN :

1661-7215

Publisher :

European Mathematical Society Publishing House

Volume :

Issue :

Pages :

1259 - 1323

Peer reviewed :

Peer reviewed

Data Set :

https://doi.org/10.4171/ggd/726

Available on ORBi :

since 15 January 2024

Statistics

Number of views

2 (0 by ULiège)

Number of downloads

1 (0 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

Bibliography

M. Baake, Á. Bustos, C. Huck, M. Lemańczyk, and A. Nickel, Number-theoretic positive entropy shifts with small centralizer and large normalizer. Ergodic Theory Dynam. Systems 41 (2021), no. 11, 3201–3226 Zbl 1486.37007 MR 4321707
M. Baake and J. A. G. Roberts, The structure of reversing symmetry groups. Bull. Austral. Math. Soc. 73 (2006), no. 3, 445–459 Zbl 1101.37030 MR 2230652
M. Baake, J. A. G. Roberts, and R. Yassawi, Reversing and extended symmetries of shift spaces. Discrete Contin. Dyn. Syst. 38 (2018), no. 2, 835–866 Zbl 1377.37027 MR 3721878
V. Berthé, W. Steiner, J. M. Thuswaldner, and R. Yassawi, Recognizability for sequences of morphisms. Ergodic Theory Dynam. Systems 39 (2019), no. 11, 2896–2931 Zbl 1476.37022 MR 4015135
S. Bezuglyi and K. Medynets, Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems. Colloq. Math. 110 (2008), no. 2, 409–429 Zbl 1142.37011 MR 2353913
M. Boyle and D. Lind, Expansive subdynamics. Trans. Amer. Math. Soc. 349 (1997), no. 1, 55–102 Zbl 0863.54034 MR 1355295
Á. Bustos, Extended symmetry groups of multidimensional subshifts with hierarchical structure. Discrete Contin. Dyn. Syst. 40 (2020), no. 10, 5869–5895 Zbl 1450.37010 MR 4128331
Á. Bustos, D. Luz, and N. Mañibo, Admissible reversing and extended symmetries for bijective substitutions. Discrete Comput. Geom. 69 (2023), no. 3, 800–833 Zbl 07661299 MR 4555870
M. I. Cortez, Zd Toeplitz arrays. Discrete Contin. Dyn. Syst. 15 (2006), no. 3, 859–881 Zbl 1130.54016 MR 2220753
M. I. Cortez and S. Petite, G-odometers and their almost one-to-one extensions. J. Lond. Math. Soc. (2) 78 (2008), no. 1, 1–20 Zbl 1154.54024 MR 2427048
V. Cyr and B. Kra, Nonexpansive Z2-subdynamics and Nivat’s conjecture. Trans. Amer. Math. Soc. 367 (2015), no. 9, 6487–6537 Zbl 1353.37035 MR 3356945
F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), no. 3, 221–239 Zbl 0348.54034 MR 461470
S. Donoso, F. Durand, A. Maass, and S. Petite, On automorphism groups of low complexity subshifts. Ergodic Theory Dynam. Systems 36 (2016), no. 1, 64–95 Zbl 1354.37024 MR 3436754
T. Downarowicz and A. Maass, Finite-rank Bratteli–Vershik diagrams are expansive. Ergodic Theory Dynam. Systems 28 (2008), no. 3, 739–747 MR 2422014
F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergodic Theory Dynam. Systems 20 (2000), no. 4, 1061–1078 Zbl 0965.37013 MR 1779393
F. Durand and J. Leroy, The constant of recognizability is computable for primitive morphisms. J. Integer Seq. 20 (2017), no. 4, article no. 17.4.5 Zbl 1359.68239 MR 3622264
F. Durand and J. Leroy, Decidability of isomorphism and factorization between minimal substitution subshifts. Discrete Anal. (2022), article no. 7 MR 4472646
A. Ehrenfeucht and G. Rozenberg, Simplifications of homomorphisms. Inform. and Control 38 (1978), no. 3, 298–309 Zbl 0387.68062 MR 509555
I. Fagnot, Sur les facteurs des mots automatiques. Theoret. Comput. Sci. 172 (1997), no. 1-2, 67–89 Zbl 0983.68102 MR 1432857
N. P. Fogg, Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Math. 1794, Springer, Berlin, 2002 MR 1970385
N. P. Frank, Multidimensional constant-length substitution sequences. Topology Appl. 152 (2005), no. 1-2, 44–69 Zbl 1074.37016 MR 2160805
N. P. Frank and N. Mañibo, Spectral theory of spin substitutions. Discrete Contin. Dyn. Syst. 42 (2022), no. 11, 5399–5435 Zbl 1505.37025 MR 4483998
F. Gähler, A. Julien, and J. Savinien, Combinatorics and topology of the Robinson tiling. C. R. Math. Acad. Sci. Paris 350 (2012), no. 11-12, 627–631 Zbl 1248.37023 MR 2956156
G. R. Goodson, Inverse conjugacies and reversing symmetry groups. Amer. Math. Monthly 106 (1999), no. 1, 19–26 Zbl 0986.20001 MR 1674141
W. H. Gottschalk, Substitution minimal sets. Trans. Amer. Math. Soc. 109 (1963), 467–491 Zbl 0121.18002 MR 190915
P. Guillon, J. Kari, and C. Zinoviadis, Determinism in subshifts. Unpublished manuscript, 2015
P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. II. Ann. of Math. (2) 43 (1942), 332–350 Zbl 0063.01888 MR 6617
G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3 (1969), 320–375 Zbl 0182.56901 MR 259881
B. Host and F. Parreau, Homomorphismes entre systèmes dynamiques définis par substitutions. Ergodic Theory Dynam. Systems 9 (1989), no. 3, 469–477 Zbl 0662.58027 MR 1016665
I. Kirat and I. Kocyigit, Remarks on self-affine fractals with polytope convex hulls. Fractals 18 (2010), no. 4, 483–498 Zbl 1209.28014 MR 2741287
J. J. Leader, Limit orbits of a power iteration for dominant eigenvalue problems. Appl. Math. Lett. 4 (1991), no. 4, 41–44 Zbl 0724.65030 MR 1117767
J.-Y. Lee, R. V. Moody, and B. Solomyak, Consequences of pure point diffraction spectra for multiset substitution systems. Discrete Comput. Geom. 29 (2003), no. 4, 525–560 Zbl 1055.37019 MR 1976605
M. Lemańczyk and M. K. Mentzen, On metric properties of substitutions. Compositio Math. 65 (1988), no. 3, 241–263 Zbl 0696.28009 MR 932072
B. Mossé, Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci. 99 (1992), no. 2, 327–334 Zbl 0763.68049 MR 1168468
C. Müllner and R. Yassawi, Automorphisms of automatic shifts. Ergodic Theory Dynam. Systems 41 (2021), no. 5, 1530–1559 Zbl 1461.37019 MR 4240609
R. Penrose, The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10 (1974), 266–271
M. Queffélec, Substitution dynamical systems—spectral analysis. Second edn., Lecture Notes in Math. 1294, Springer, Berlin, 2010 Zbl 1225.11001 MR 2590264
E. A. Robinson, Jr., On the table and the chair. Indag. Math. (N.S.) 10 (1999), no. 4, 581–599 Zbl 1024.37009 MR 1820555
R. T. Rockafellar, Convex analysis. Princeton Mat. Ser., No. 28, Princeton University Press, Princeton, NJ, 1970 Zbl 0193.18401 MR 0274683
V. Salo and I. Törmä, Block maps between primitive uniform and Pisot substitutions. Ergodic Theory Dynam. Systems 35 (2015), no. 7, 2292–2310 Zbl 1352.37039 MR 3394119
B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20 (1998), no. 2, 265–279 Zbl 0919.52017 MR 1637896
B. Solomyak, Eigenfunctions for substitution tiling systems. In Probability and number theory—Kanazawa 2005, pp. 433–454, Adv. Stud. Pure Math. 49, Mathematical Society of Japan, Tokyo, 2007 Zbl 1139.37009 MR 2405614
R. S. Strichartz and Y. Wang, Geometry of self-affine tiles. I. Indiana Univ. Math. J. 48 (1999), no. 1, 1–23 Zbl 0938.52017 MR 1722192
A. Vince, Digit tiling of Euclidean space. In Directions in mathematical quasicrystals, pp.
329–370, CRM Monogr. Ser. 13, American Mathematical Society, Providence, RI, 2000 Zbl 0972.52012 MR 1798999