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SOLENOIDAL MAPS, AUTOMATIC SEQUENCES, VAN DER PUT SERIES, AND MEALY AUTOMATA

Published online by Cambridge University Press:  06 April 2022

ROSTISLAV GRIGORCHUK
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX77843-3368, USA e-mail: grigorch@math.tamu.edu
DMYTRO SAVCHUK*
Affiliation:
Department of Mathematics and Statistics, University of South Florida, 4202 E Fowler Ave, Tampa, FL33620-5700, USA
*

Abstract

The ring $\mathbb Z_{d}$ of d-adic integers has a natural interpretation as the boundary of a rooted d-ary tree $T_{d}$. Endomorphisms of this tree (that is, solenoidal maps) are in one-to-one correspondence with 1-Lipschitz mappings from $\mathbb Z_{d}$ to itself. In the case when $d=p$ is prime, Anashin [‘Automata finiteness criterion in terms of van der Put series of automata functions’,p-Adic Numbers Ultrametric Anal. Appl.4(2) (2012), 151–160] showed that $f\in \mathrm {Lip}^{1}(\mathbb Z_{p})$ is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a p-automatic sequence over a finite subset of $\mathbb Z_{p}\cap \mathbb Q$. We generalize this result to arbitrary integers $d\geq 2$ and describe the explicit connection between the Moore automaton producing such a sequence and the Mealy automaton inducing the corresponding endomorphism of a rooted tree. We also produce two algorithms converting one automaton to the other and vice versa. As a demonstration, we apply our algorithms to the Thue–Morse sequence and to one of the generators of the lamplighter group acting on the binary rooted tree.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Ben Martin

The first author graciously acknowledges support from the Simons Foundation through Collaboration Grant No. 527814 and is also supported by the mega-grant of the Russian Federation Government (N14.W03.31.0030). He also gratefully acknowledges support of the Swiss National Science Foundation.

The second author greatly appreciates the support of the Simons Foundation through Collaboration Grant No. 317198.

References

Ahmed, E. and Savchuk, D., ‘Endomorphisms of regular rooted trees induced by the action of polynomials on the ring ${\mathbb{Z}}_d$ of $d$-adic integers’, J. Algebra Appl. 19 (2020), 2050154.CrossRefGoogle Scholar
Allouche, J.-P. and Shallit, J., Automatic Sequences: Theory, Applications, Generalizations (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
Anashin, V., ‘Ergodic transformations in the space of $p$-adic integers’, in: $p$-Adic Mathematical Physics, AIP Conference Proceedings, 826 (eds. Yu. Khrennikov, Z. Rakić and I.Volovich) (American Institute of Physics, Melville, NY, 2006), 324.Google Scholar
Anashin, V., ‘Automata finiteness criterion in terms of van der Put series of automata functions’, p-Adic Numbers Ultrametric Anal. Appl. 4(2) (2012), 151160.CrossRefGoogle Scholar
Anashin, V. S., Khrennikov, A. Y. and Yurova, E. I., ‘Characterization of ergodic $p$-adic dynamical systems in terms of the van der Put basis’, Dokl. Akad. Nauk 438(2) (2011), 151153 [English translation: Dokl. Math. 83(3) (2011), 306–308].Google Scholar
Bartholdi, L., Grigorchuk, R. and Nekrashevych, V., ‘From fractal groups to fractal sets’, in: Fractals in Graz 2001, Trends in Mathematics (eds P. Grabner and W. Woess) (Birkhäuser, Basel, 2003), 25118.CrossRefGoogle Scholar
Bartholdi, L. and Grigorchuk, R. I., ‘On the spectrum of Hecke type operators related to some fractal groups’, Tr. Mat. Inst. Steklova 231(Din. Sist., Avtom. i Beskon. Gruppy) (2000), 545.Google Scholar
Bartholdi, L. I. and Nekrashevych, V. V., ‘Thurston equivalence of topological polynomials’, Acta Math. 197(1) (2006), 151.CrossRefGoogle Scholar
Bartholdi, L. I. and Šuniḱ, Z., ‘Some solvable automaton groups’, in: Topological and Asymptotic Aspects of Group Theory, Contemporary Mathematics, 394 (eds. R. Grigorchuk, M. Mihalik, M. Sapir and Z. $\check{\rm S}\text{uni}\acute{\rm k}$) (American Mathematical Society, Providence, RI, 2006), 1129.CrossRefGoogle Scholar
Bernstein, D. J. and Lagarias, J. C., ‘The $3x+1$ conjugacy map’, Canad. J. Math. 48(6) (1996), 11541169.CrossRefGoogle Scholar
Cain, A. J., ‘Automaton semigroups’, Theoret. Comput. Sci. 410(47–49) (2009), 50225038.CrossRefGoogle Scholar
Cull, P. and Nelson, I., ‘Error-correcting codes on the towers of Hanoi graphs’, Discrete Math. 208/209 (1999), 157175. Combinatorics (Assisi, 1996).CrossRefGoogle Scholar
Garzon, M. and Zalcstein, Y., ‘The complexity of Grigorchuk groups with application to cryptography,’ Theoret. Comput. Sci. 88(1) (1991), 8398.CrossRefGoogle Scholar
Goresky, M. and Klapper, A., Algebraic Shift Register Sequences (Cambridge University Press, Cambridge, 2012).Google Scholar
Grigorchuk, R., Lenz, D. and Nagnibeda, T., ‘Spectra of Schreier graphs of Grigorchuk’s group and Schroedinger operators with aperiodic order’, Math. Ann. 370(3–4) (2018), 16071637.CrossRefGoogle Scholar
Grigorchuk, R., Leonov, Y., Nekrashevych, V. and Sushchansky, V., ‘Self-similar groups, automatic sequences, and unitriangular representations’, Bull. Math. Sci. 6(2) (2016), 231285.CrossRefGoogle Scholar
Grigorchuk, R. and Šunić, Z., ‘Self-similarity and branching in group theory’, in: Groups St. Andrews 2005, I, London Mathematical Society Lecture Note Series, 339 (eds. C. M. Campbell, M. R. Quick, E. F. Robertson and G. C. Smith) (Cambridge University Press, Cambridge, 2007), 3695.CrossRefGoogle Scholar
Grigorchuk, R. and Šunić, Z., ‘Schreier spectrum of the Hanoi Towers group on three pegs’, in: Analysis on Graphs and Its Applications, Proceedings of Symposia in Pure Mathematics, 77 (eds P. Exner, J. Keating, P. Kuchment, T. Sunada and A. Teplyaev) (American Mathematical Society, Providence, RI, 2008), 183198.CrossRefGoogle Scholar
Grigorchuk, R. and Šuniḱ, Z., ‘Asymptotic aspects of Schreier graphs and Hanoi Towers groups’, C. R. Math. Acad. Sci. Paris 342(8) (2006), 545550.CrossRefGoogle Scholar
Grigorchuk, R. I., ‘On Burnside’s problem on periodic groups’, Funktsional. Anal. i Prilozhen. 14(1) (1980), 5354.CrossRefGoogle Scholar
Grigorchuk, R. I., ‘On the Milnor problem of group growth’, Dokl. Akad. Nauk SSSR 271(1) (1983), 3033.Google Scholar
Grigorchuk, R. I., Linnell, P., Schick, T. and Żuk, A., ‘On a question of Atiyah’, C. R. Acad. Sci. Paris Sér. I Math. 331(9) (2000), 663668.CrossRefGoogle Scholar
Grigorchuk, R. I., Nekrashevich, V. V. and Sushchanskiĭ, V. I., ‘Automata, dynamical systems, and groups’, Tr. Mat. Inst. Steklova 231(Din. Sist., Avtom. i Beskon. Gruppy) (2000), 134214.Google Scholar
Grigorchuk, R. I. and Żuk, A., ‘The lamplighter group as a group generated by a 2-state automaton, and its spectrum’, Geom. Dedicata 87(1–3) (2001), 209244.CrossRefGoogle Scholar
Grigorchuk, R. I. and Żuk, A., ‘On a torsion-free weakly branch group defined by a three state automaton’, Internat. J. Algebra Comput. 12(1–2) (2002), 223246.CrossRefGoogle Scholar
Katok, S., $p$-Adic Analysis Compared with Real, Student Mathematical Library, 37 (American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2007).CrossRefGoogle Scholar
Larin, M. V., ‘Transitive polynomial transformations of residue rings’, Diskret. Mat. 14(2) (2002), 2032.Google Scholar
Mahler, K., $p$-Adic Numbers and Their Functions, 2nd edn, Cambridge Tracts in Mathematics, 76 (Cambridge University Press, Cambridge–New York, 1981).Google Scholar
Miasnikov, A. and Savchuk, D., ‘An example of an automatic graph of intermediate growth’, Ann. Pure Appl. Logic 166(10) (2015), 10371048.CrossRefGoogle Scholar
Miasnikov, A. and Šunić, Z., ‘Cayley graph automatic groups are not necessarily Cayley graph biautomatic’, in: Language and Automata Theory and Applications, Lecture Notes in Computer Science, 7183 (eds A.-H. Dediu and C. Martín-Vide) (Springer, Heidelberg, 2012), 401407.CrossRefGoogle Scholar
Myasnikov, A., Shpilrain, V. and Ushakov, A., Non-Commutative Cryptography and Complexity of Group-Theoretic Problems, Mathematical Surveys and Monographs, 177 (American Mathematical Society, Providence, RI, 2011). With an appendix by Natalia Mosina.CrossRefGoogle Scholar
Myasnikov, A. G. and Ushakov, A., ‘Random subgroups and analysis of the length-based and quotient attacks’, J. Math. Cryptol. 2(1) (2008), 2961.CrossRefGoogle Scholar
Nekrashevych, V., Self-Similar Groups, Mathematical Surveys and Monographs, 117 (American Mathematical Society, Providence, RI, 2005).CrossRefGoogle Scholar
Nekrashevych, V. and Sidki, S., ‘Automorphisms of the binary tree: state-closed subgroups and dynamics of $1/ 2$-endomorphisms’, in: Groups: Topological, Combinatorial and Arithmetic Aspects, London Mathematical Society Lecture Note Series, 311 (ed. T. Müller) (Cambridge University Press, Cambridge, 2004), 375404.CrossRefGoogle Scholar
Petrides, G., ‘Cryptanalysis of the public key cryptosystem based on the word problem on the Grigorchuk groups’, in: Cryptography and Coding, Lecture Notes in Computer Science, 2898 (ed. K. Paterson) (Springer, Berlin, 2003), 234244.CrossRefGoogle Scholar
Schikhof, W. H.. Ultrametric Calculus: An Introduction to p-Adic Analysis, Cambridge Studies in Advanced Mathematics, 4 (Cambridge University Press, Cambridge, 1984).Google Scholar
Sholomov, L. A., Foundations of the Theory of Discrete Logical and Computational Devices (in Russian) (Nauka, Moscow, 1980).Google Scholar
Walsh, J. L., ‘A closed set of normal orthogonal functions’, Amer. J. Math. 45(1) (1923), 524.CrossRefGoogle Scholar

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