Abstract
We present a survey of results obtained on symbolic dynamical systems called dendric shifts. We state and sketch the proofs (sometimes new ones) of the main results obtained on these shifts. This includes the Return Theorem and the Finite Index Basis Theorem which both put in evidence the central role played by free groups in these systems. We also present a series of applications of these results, including some on profinite semigroups and some on dimension groups.
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References
Almeida, J.: Profinite groups associated with weakly primitive substitutions. Fundam. Prikl. Mat. 11(3), 13–48 (2005)
Almeida, J., Costa, A.: A geometric interpretation of the Schützenberger group of a minimal subshift. Ark. Mat. 54(2), 243–275 (2016)
Arnoux, P., Rauzy, G.: Représentation géométrique de suites de complexité \(2n+1\). Bull. Soc. Math. France 119(2), 199–215 (1991)
Balková, L., Pelantová, E., Steiner, W.: Sequences with constant number of return words. Monatsh. Math. 155(3–4), 251–263 (2008)
Berstel, J., De Felice, C., Perrin, D., Reutenauer, C., Rindone, G.: Bifix codes and Sturmian words. J. Algebra 369, 146–202 (2012)
Berthé, V., et al.: Dimension groups of dendric shifts (2018, preprint)
Berthé, V., et al.: Acyclic, connected and tree sets. Monatsh. Math. 176(4), 521–550 (2015)
Berthé, V., et al.: The finite index basis property. J. Pure Appl. Algebra 219, 2521–2537 (2015)
Berthé, V., et al.: Maximal bifix decoding. Discrete Math. 338(5), 725–742 (2015)
Berthé, V., Rigo, M. (eds.): Combinatorics, Automata and Number Theory. Encyclopedia of Mathematics and its Applications, vol. 135. Cambridge University Press, Cambridge (2010)
Davidson, K.R.: \(C^*\)-Algebras by Example. Fields Institute Monographs, vol. 6. American Mathematical Society, Providence (1996)
Dolce, F., Perrin, D.: Neutral and tree sets of arbitrary characteristic. Theor. Comput. Sci. 658, 159–174 (2017)
Durand, F.: A characterization of substitutive sequences using return words. Discrete Math. 179(1–3), 89–101 (1998)
Durand, F., Host, B., Skau, C.: Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergod. Theory Dyn. Syst. 19(4), 953–993 (1999)
Fogg, N.P.: Substitutions in Dynamics, Arithmetics and Combinatorics. LNM, vol. 1794. Springer, Berlin (2002). https://doi.org/10.1007/b13861. Berthé, V., Ferenczi, S., Mauduit, C., Siegel, A. (eds.)
Giordano, T., Putnam, I.F., Skau, C.F.: Topological orbit equivalence and \(C^*\)-crossed products. J. Reine Angew. Math. 469, 51–111 (1995)
Justin, J., Vuillon, L.: Return words in Sturmian and episturmian words. Theor. Inf. Appl. 34(5), 343–356 (2000)
Tan, B., Wen, Z.-X., Zhang, Y.: The structure of invertible substitutions on a three-letter alphabet. Adv. Appl. Math. 32(4), 736–753 (2004)
Acknowledgement
The results presented in this paper have been obtained through a cooperation with a group including Valérie Berthé, Paulina Cecchi, Clelia De Felice, Vincent Delecroix, Francesco Dolce, Fabien Durand, Julien Leroy, Samuel Petite, Christophe Reutenauer and Giuseppina Rindone. They are gratefully acknowledged for their help in the preparation.
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Perrin, D. (2018). Groups, Languages and Dendric Shifts. In: Hoshi, M., Seki, S. (eds) Developments in Language Theory. DLT 2018. Lecture Notes in Computer Science(), vol 11088. Springer, Cham. https://doi.org/10.1007/978-3-319-98654-8_5
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DOI: https://doi.org/10.1007/978-3-319-98654-8_5
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