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Growth degree classification for finitely generated semigroups of integer matrices

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Abstract

Let \({\mathcal {A}}\) be a finite set of \(d\times d\) matrices with integer entries and let \(m_n({\mathcal {A}})\) be the maximum norm of a product of \(n\) elements of \({\mathcal {A}}\). In this paper, we classify gaps in the growth of \(m_n({\mathcal {A}})\); specifically, we prove that \(\lim _{n\rightarrow \infty } \log m_n({\mathcal {A}})/\log n\in \mathbb {Z}_{\geqslant 0}\cup \{\infty \}.\) This has applications to the growth of regular sequences as defined by Allouche and Shallit.

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References

  1. Allouche, J.-P., Shallit, J.: The ring of \(k\)-regular sequences. Theor. Comput. Sci. 98(2), 163–197 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bell, J.P.: A gap result for the norms of semigroups of matrices. Linear Algebra Appl. 402, 101–110 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bell, J.P., Coons, M., Hare, K.G.: The minimal growth of a \(k\)-regular sequence. Bull. Aust. Math. Soc. 90(2), 195–203 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berstel, J., Reutenauer, C.: Noncommutative Rational Series with Applications, Encyclopedia of Mathematics and Its Applications, vol. 137. Cambridge University Press, Cambridge (2011)

    Google Scholar 

  5. Blondel, V.D., Nesterov, Y.: Computationally efficient approximations of the joint spectral radius. SIAM J. Matrix Anal. Appl. 27(1), 256–272 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Blondel, V.D., Theys, J., Vladimirov, A.A.: An elementary counterexample to the finiteness conjecture. SIAM J. Matrix Anal. Appl. 24(4), 963–970 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bousch, T., Mairesse, J.: Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture. J. Am. Math. Soc. 15(1), 77–111 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cicone, A., Guglielmi, N., Serra-Capizzano, S., Zennaro, M.: Finiteness property of pairs of \(2\times 2\) sign-matrices via real extremal polytope norms. Linear Algebra Appl. 432(2–3), 796–816 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dummit, D.S., Foote, R.M.: Abstract Algebra, 3rd edn. Wiley, Hoboken, NJ (2004)

    MATH  Google Scholar 

  10. Hare, K.G., Morris, I.D., Sidorov, N., Theys, J.: An explicit counterexample to the Lagarias–Wang finiteness conjecture. Adv. Math. 226(6), 4667–4701 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hare, K.G., Morris, I.D., Sidorov, N.: Extremal sequences of polynomial complexity. Math. Proc. Camb. Philos. Soc. 155(2), 191–205 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Heil, C., Strang, G.: Continuity of the joint spectral radius: application to wavelets. In: Bojanczyk, A., Cybenko, G. (eds.) Linear Algebra for Signal Processing (Minneapolis, MN, 1992), IMA Volumes in Mathematics and Its Applications, vol. 69, pp. 51–61. Springer, New York (1995)

  13. Jacobson, N.: Structure theory of simple rings without finiteness assumptions. Trans. Am. Math. Soc. 57, 228–245 (1945)

    Article  MATH  Google Scholar 

  14. Jungers, R.: The joint spectral radius. In: Lecture Notes in Control and Information Sciences, vol. 385. Springer, Berlin (2009)

  15. Jungers, R.M., Blondel, V.D.: On the finiteness property for rational matrices. Linear Algebra Appl. 428(10), 2283–2295 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jungers, R.M., Protasov, V., Blondel, V.D.: Efficient algorithms for deciding the type of growth of products of integer matrices. Linear Algebra Appl. 428(10), 2296–2311 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kozyakin, V.S.: A dynamical systems construction of a counterexample to the finiteness conjecture. In: Proceedings of the 44th IEEE Conference on Decision and Control, European Control Conference, pp. 2338–2343 (2005)

  18. Kronecker, L.: Zwei Sätze über Gleichungen mit ganzzahligen coefficienten. J. Reine Angew. Math. 53, 173–175 (1857)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lagarias, J.C., Wang, Y.: The finiteness conjecture for the generalized spectral radius of a set of matrices. Linear Algebra Appl. 214, 17–42 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rota, G.-C., Strang, G.: A note on the joint spectral radius. Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22, 379–381 (1960)

    Article  MathSciNet  Google Scholar 

  21. Theys, J.: Joint Spectral Radius: Theory and Approximations. PhD Thesis, Université Catholique de Louvin (2005)

  22. Vladimir, Y.P., Raphaël, M.J.: Resonance and marginal instability of switching systems. Nonlinear Anal. Hybrid Syst. 17, 81–93 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We thank Vladimir Protasov for bringing our attention to his current result with Jungers [22] as well as providing us with a preprint of that work. Research of J. P. Bell was supported by NSERC Grant 326532-2011, the research of M. Coons was supported by ARC Grant DE140100223, and the research of K. G. Hare was supported by NSERC Grant RGPIN-2014-03154.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, Canada

    Jason P. Bell & Kevin G. Hare

  2. School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, Australia

    Michael Coons

Authors
  1. Jason P. Bell
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  2. Michael Coons
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  3. Kevin G. Hare
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Correspondence to Jason P. Bell.

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Communicated by Jean-Eric Pin.

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Bell, J.P., Coons, M. & Hare, K.G. Growth degree classification for finitely generated semigroups of integer matrices. Semigroup Forum 92, 23–44 (2016). https://doi.org/10.1007/s00233-015-9725-1

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  • DOI: https://doi.org/10.1007/s00233-015-9725-1

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