{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:18:19Z","timestamp":1758824299737,"version":"3.37.3"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,5,11]],"date-time":"2022-05-11T00:00:00Z","timestamp":1652227200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,5,11]],"date-time":"2022-05-11T00:00:00Z","timestamp":1652227200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"deutsche forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB 1283\/2 2021-317210226"],"award-info":[{"award-number":["SFB 1283\/2 2021-317210226"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2023,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we deal with reversing and extended symmetries of subshifts generated by bijective substitutions. We survey some general algebraic and dynamical properties of these subshifts and recall known results regarding their symmetry groups. We provide equivalent conditions for a permutation on the alphabet to generate a reversing\/extended symmetry, and algorithms how to compute them. Moreover, for any finite group <jats:italic>H<\/jats:italic> and any subgroup <jats:italic>P<\/jats:italic> of the <jats:italic>d<\/jats:italic>-dimensional hyperoctahedral group, we construct a bijective substitution which generates an aperiodic subshift with symmetry group <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {Z}}^{d}\\times H$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>Z<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>d<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mi>H<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and extended symmetry group <jats:inline-formula><jats:alternatives><jats:tex-math>$$({\\mathbb {Z}}^{d} \\rtimes P)\\times H$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>Z<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>d<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>\u22ca<\/mml:mo>\n                    <mml:mi>P<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mi>H<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. A similar construction with the same symmetry group, but with extended symmetry group <jats:inline-formula><jats:alternatives><jats:tex-math>$$({\\mathbb {Z}}^{d} \\times H) \\rtimes P$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>Z<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>d<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>\u22ca<\/mml:mo>\n                    <mml:mi>P<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is also provided under a mild assumption on the dimension.<\/jats:p>","DOI":"10.1007\/s00454-022-00387-8","type":"journal-article","created":{"date-parts":[[2022,5,11]],"date-time":"2022-05-11T19:25:15Z","timestamp":1652297115000},"page":"800-833","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Admissible Reversing and Extended Symmetries for Bijective Substitutions"],"prefix":"10.1007","volume":"69","author":[{"given":"\u00c1lvaro","family":"Bustos","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Luz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Neil","family":"Ma\u00f1ibo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,5,11]]},"reference":[{"key":"387_CR1","doi-asserted-by":"crossref","unstructured":"Baake, M.: A brief guide to reversing and extended symmetries of dynamical systems. In: Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol. 2213, pp. 117\u2013135. Springer, Cham (2018)","DOI":"10.1007\/978-3-319-74908-2_9"},{"issue":"11","key":"387_CR2","doi-asserted-by":"publisher","first-page":"3171","DOI":"10.1063\/1.526087","volume":"25","author":"M Baake","year":"1984","unstructured":"Baake, M.: Structure and representations of the hyperoctahedral group. J. Math. Phys. 25(11), 3171\u20133182 (1984)","journal-title":"J. Math. Phys."},{"issue":"11","key":"387_CR3","doi-asserted-by":"publisher","first-page":"3201","DOI":"10.1017\/etds.2020.111","volume":"41","author":"M Baake","year":"2021","unstructured":"Baake, M., Bustos, \u00c1., Huck, C., Lema\u0144czyk, M., Nickel, A.: Number-theoretic positive entropy shifts with small centralizer and large normalizer. Ergodic Theory Dyn. Syst. 41(11), 3201\u20133226 (2021)","journal-title":"Ergodic Theory Dyn. Syst."},{"key":"387_CR4","volume-title":"Aperiodic Order, vol. 1. A Mathematical Invitation. Encyclopedia of Mathematics and Its Applications","author":"M Baake","year":"2013","unstructured":"Baake, M., Grimm, U.: Aperiodic Order, vol. 1. A Mathematical Invitation. Encyclopedia of Mathematics and Its Applications, vol. 149. Cambridge University Press, Cambridge (2013)"},{"issue":"3","key":"387_CR5","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1017\/S0004972700035450","volume":"73","author":"M Baake","year":"2006","unstructured":"Baake, M., Roberts, J.A.G.: The structure of reversing symmetry groups. Bull. Aust. Math. Soc. 73(3), 445\u2013459 (2006)","journal-title":"Bull. Aust. Math. Soc."},{"issue":"2","key":"387_CR6","doi-asserted-by":"publisher","first-page":"835","DOI":"10.3934\/dcds.2018036","volume":"38","author":"M Baake","year":"2018","unstructured":"Baake, M., Roberts, J.A.G., Yassawi, R.: Reversing and extended symmetries of shift spaces. Discrete Contin. Dyn. Syst. 38(2), 835\u2013866 (2018)","journal-title":"Discrete Contin. Dyn. Syst."},{"issue":"4","key":"387_CR7","doi-asserted-by":"publisher","first-page":"1289","DOI":"10.1017\/etds.2016.66","volume":"38","author":"A Bartlett","year":"2018","unstructured":"Bartlett, A.: Spectral theory of $$\\mathbb{Z}^d$$ substitutions. Ergodic Theory Dyn. Syst. 38(4), 1289\u20131341 (2018)","journal-title":"Ergodic Theory Dyn. Syst."},{"issue":"2","key":"387_CR8","doi-asserted-by":"publisher","first-page":"409","DOI":"10.4064\/cm110-2-6","volume":"110","author":"S Bezuglyi","year":"2008","unstructured":"Bezuglyi, S., Medynets, K.: Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems. Colloq. Math. 110(2), 409\u2013429 (2008)","journal-title":"Colloq. Math."},{"issue":"1","key":"387_CR9","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1090\/S0002-9947-1988-0927684-2","volume":"306","author":"M Boyle","year":"1988","unstructured":"Boyle, M., Lind, D., Rudolph, D.: The automorphism group of a shift of finite type. Trans. Am. Math. Soc. 306(1), 71\u2013114 (1988)","journal-title":"Trans. Am. Math. Soc."},{"issue":"10","key":"387_CR10","doi-asserted-by":"publisher","first-page":"5869","DOI":"10.3934\/dcds.2020250","volume":"40","author":"\u00c1 Bustos","year":"2020","unstructured":"Bustos, \u00c1.: Extended symmetry groups of multidimensional subshifts with hierarchical structure. Discrete Contin. Dyn. Syst. 40(10), 5869\u20135895 (2020)","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"387_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-14034-1","volume-title":"Cellular Automata and Groups. Springer Monographs in Mathematics","author":"T Ceccherini-Silberstein","year":"2010","unstructured":"Ceccherini-Silberstein, T., Coornaert, M.: Cellular Automata and Groups. Springer Monographs in Mathematics. Springer, Berlin (2010)"},{"issue":"4","key":"387_CR12","doi-asserted-by":"publisher","first-page":"622","DOI":"10.1007\/s00454-008-9108-4","volume":"40","author":"MI Cortez","year":"2008","unstructured":"Cortez, M.I., Durand, F.: Self-similar tiling systems, topological factors and stretching factors. Discrete Comput. Geom. 40(4), 622\u2013640 (2008)","journal-title":"Discrete Comput. Geom."},{"issue":"5","key":"387_CR13","doi-asserted-by":"publisher","first-page":"2891","DOI":"10.3934\/dcds.2020153","volume":"40","author":"MI Cortez","year":"2020","unstructured":"Cortez, M.I., Petite, S.: Realization of big centralizers of minimal aperiodic actions on the Cantor set. Discrete Contin. Dyn. Syst. 40(5), 2891\u20132901 (2020)","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"387_CR14","doi-asserted-by":"crossref","unstructured":"Coven, E.M.: Endomorphisms of substitution minimal sets. Z.\u00a0Wahrscheinlichkeitstheorie und Verw. Gebiete 20, 129\u2013133 (1971\/72)","DOI":"10.1007\/BF00536290"},{"key":"387_CR15","unstructured":"Coven, E.M., Quas, A., Yassawi, R.: Computing automorphism groups of shifts using atypical equivalence classes. Discrete Anal. 2016, #\u00a03 (2016)"},{"issue":"2","key":"387_CR16","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1090\/proc12719","volume":"144","author":"V Cyr","year":"2016","unstructured":"Cyr, V., Kra, B.: The automorphism group of a shift of subquadratic growth. Proc. Am. Math. Soc. 144(2), 613\u2013621 (2016)","journal-title":"Proc. Am. Math. Soc."},{"issue":"1","key":"387_CR17","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1017\/etds.2015.70","volume":"36","author":"S Donoso","year":"2016","unstructured":"Donoso, S., Durand, F., Maass, A., Petite, S.: On automorphism groups of low complexity subshifts. Ergodic Theory Dyn. Syst. 36(1), 64\u201395 (2016)","journal-title":"Ergodic Theory Dyn. Syst."},{"key":"387_CR18","unstructured":"Durand, F., Leroy, J.: Decidability of the isomorphism and the factorization between minimal substitution subshifts (2018). arXiv:1806.04891"},{"issue":"4","key":"387_CR19","doi-asserted-by":"publisher","first-page":"1105","DOI":"10.1016\/j.indag.2018.05.011","volume":"29","author":"R Fokkink","year":"2018","unstructured":"Fokkink, R., Yassawi, R.: Topological rigidity of linear cellular automaton shifts. Indag. Math. 29(4), 1105\u20131113 (2018)","journal-title":"Indag. Math."},{"issue":"2","key":"387_CR20","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1017\/S0143385702001256","volume":"23","author":"NP Frank","year":"2003","unstructured":"Frank, N.P.: Substitution sequences in $$\\mathbb{Z}^d$$ with a non-simple Lebesgue component in the spectrum. Ergodic Theory Dyn. Syst. 23(2), 519\u2013532 (2003)","journal-title":"Ergodic Theory Dyn. Syst."},{"issue":"1\u20132","key":"387_CR21","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1016\/j.topol.2004.08.014","volume":"152","author":"NP Frank","year":"2005","unstructured":"Frank, N.P.: Multidimensional constant-length substitution sequences. Topol. Appl. 152(1\u20132), 44\u201369 (2005)","journal-title":"Topol. Appl."},{"key":"387_CR22","doi-asserted-by":"crossref","unstructured":"Frank, N.P.: Introduction to hierarchical tiling dynamical systems. In: Substitution and Tiling Dynamics. Introduction to Self-Inducing Structures. Lecture Notes in Math., vol. 2273, pp. 33\u201395. Springer, Cham (2020)","DOI":"10.1007\/978-3-030-57666-0_2"},{"issue":"1","key":"387_CR23","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1080\/00029890.1999.12005002","volume":"106","author":"GR Goodson","year":"1999","unstructured":"Goodson, G.R.: Inverse conjugacies and reversing symmetry groups. Am. Math. Monthly 106(1), 19\u201326 (1999)","journal-title":"Am. Math. Monthly"},{"key":"387_CR24","volume-title":"The Theory of Groups","author":"M Hall Jr","year":"1976","unstructured":"Hall, M., Jr.: The Theory of Groups. Chelsea, New York (1976)"},{"issue":"2","key":"387_CR25","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/s00220-004-1195-3","volume":"254","author":"C Holton","year":"2005","unstructured":"Holton, C., Radin, C., Sadun, L.: Conjugacies for tiling dynamical systems. Commun. Math. Phys. 254(2), 343\u2013359 (2005)","journal-title":"Commun. Math. Phys."},{"key":"387_CR26","doi-asserted-by":"crossref","unstructured":"Kellendonk, J., Yassawi, R.: The Ellis semigroup of bijective substitutions. Groups Geom. Dyn. (2021). https:\/\/ems.press\/journals\/ggd\/articles\/3897337","DOI":"10.4171\/GGD\/640"},{"issue":"3","key":"387_CR27","doi-asserted-by":"publisher","first-page":"800","DOI":"10.1007\/s00454-019-00153-3","volume":"65","author":"S Labb\u00e9","year":"2021","unstructured":"Labb\u00e9, S.: Substitutive structure of Jeandel\u2013Rao aperiodic tilings. Discrete Comput. Geom. 65(3), 800\u2013855 (2021)","journal-title":"Discrete Comput. Geom."},{"issue":"1\u20132","key":"387_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0167-2789(97)00199-1","volume":"112","author":"JSW Lamb","year":"1998","unstructured":"Lamb, J.S.W., Roberts, J.A.G.: Time-reversal symmetry in dynamical systems: a survey. Phys. D 112(1\u20132), 1\u201339 (1998)","journal-title":"Phys. D"},{"issue":"3","key":"387_CR29","first-page":"241","volume":"65","author":"M Lema\u0144czyk","year":"1988","unstructured":"Lema\u0144czyk, M., Mentzen, M.K.: On metric properties of substitutions. Compositio Math. 65(3), 241\u2013263 (1988)","journal-title":"Compositio Math."},{"key":"387_CR30","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511626302","volume-title":"An Introduction to Symbolic Dynamics and Coding","author":"D Lind","year":"1995","unstructured":"Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)"},{"issue":"3","key":"387_CR31","doi-asserted-by":"publisher","first-page":"1086","DOI":"10.1017\/etds.2016.58","volume":"38","author":"GR Maloney","year":"2018","unstructured":"Maloney, G.R., Rust, D.: Beyond primitivity for one-dimensional substitution subshifts and tiling spaces. Ergodic Theory Dyn. Syst. 38(3), 1086\u20131117 (2018)","journal-title":"Ergodic Theory Dyn. Syst."},{"issue":"5","key":"387_CR32","doi-asserted-by":"publisher","first-page":"1530","DOI":"10.1017\/etds.2020.13","volume":"41","author":"C M\u00fcllner","year":"2021","unstructured":"M\u00fcllner, C., Yassawi, R.: Automorphisms of automatic shifts. Ergodic Theory Dyn. Syst. 41(5), 1530\u20131559 (2021)","journal-title":"Ergodic Theory Dyn. Syst."},{"key":"387_CR33","unstructured":"O\u2019Farrell, A.G., Short, I.: Reversibility in Dynamics and Group Theory. London Mathematical Society Lecture Note Series, vol. 416. Cambridge University Press, Cambridge (2015)"},{"key":"387_CR34","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-11212-6","volume-title":"Substitution Dynamical Systems\u2014Spectral Analysis. Lecture Notes in Mathematics","author":"M Queff\u00e9lec","year":"2010","unstructured":"Queff\u00e9lec, M.: Substitution Dynamical Systems\u2014Spectral Analysis. Lecture Notes in Mathematics, vol. 1294. Springer, Berlin (2010)"},{"key":"387_CR35","doi-asserted-by":"publisher","first-page":"565","DOI":"10.1090\/S0002-9904-1940-07261-1","volume":"46","author":"IE Segal","year":"1940","unstructured":"Segal, I.E.: The automorphisms of the symmetric group. Bull. Am. Math. Soc. 46, 565 (1940)","journal-title":"Bull. Am. Math. Soc."},{"issue":"3","key":"387_CR36","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1155\/S0161171289000578","volume":"12","author":"SK Sehgal","year":"1989","unstructured":"Sehgal, S.K.: On the normalizer of a group in the Cayley representation. Int. J. Math. Math. Sci. 12(3), 459\u2013462 (1989)","journal-title":"Int. J. Math. Math. Sci."},{"issue":"2","key":"387_CR37","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1007\/PL00009386","volume":"20","author":"B Solomyak","year":"1998","unstructured":"Solomyak, B.: Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20(2), 265\u2013279 (1998)","journal-title":"Discrete Comput. Geom."}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-022-00387-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-022-00387-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-022-00387-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,3]],"date-time":"2023-03-03T20:02:37Z","timestamp":1677873757000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-022-00387-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,11]]},"references-count":37,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,4]]}},"alternative-id":["387"],"URL":"https:\/\/doi.org\/10.1007\/s00454-022-00387-8","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"type":"print","value":"0179-5376"},{"type":"electronic","value":"1432-0444"}],"subject":[],"published":{"date-parts":[[2022,5,11]]},"assertion":[{"value":"26 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 July 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 December 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 May 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}