Perron-Frobenius theory and frequency convergence for reducible substitutions

Article, Preprint English OPEN
Lustig, Martin; Uyanik, Caglar;
  • Publisher: HAL CCSD
  • Related identifiers: doi: 10.3934/dcds.2017015
  • Subject: reducible substitutions | Mathematics - Rings and Algebras | [ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS] | Perron-Frobenius | primitive substitutions | Mathematics - Dynamical Systems | [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA] | invariant measures | [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] | [ MATH.MATH-RA ] Mathematics [math]/Rings and Algebras [math.RA] | invariant-measures | matrices | frequencies | Perron-Frobenius theory

v.2 Minor revisions for submission; International audience; We prove a general version of the classical Perron-Frobenius convergence property for reducible matrices. We then apply this result to reducible substitutions and use it to produce limit frequencies for factors... View more
  • References (14)
    14 references, page 1 of 2

    [AGN11] Marianne Akian, St´ephane Gaubert, and Roger Nussbaum. A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones. arXiv preprint arXiv:1112.5968, 2011.

    [BHL15] Nicolas B´edaride, Arnaud Hilion, and Martin Lustig. Invariant measures for train track towers. arXiv preprint arXiv:1503.08000, 2015.

    [BKMS10] Sergey Bezuglyi, Jan Kwiatkowski, Konstantin Medynets, and Boris Solomyak. Invariant measures on stationary Bratteli diagrams. Ergodic Theory Dynam. Systems, 30(4):973-1007, 2010.

    [BSS12] Peter Butkoviˇc, Hans Schneider, and Serge˘ı Sergeev. Z-matrix equations in max-algebra, nonnegative linear algebra and other semirings. Linear Multilinear Algebra, 60(10):1191-1210, 2012.

    [ESS14] Gernot Michael Engel, Hans Schneider, and Serge˘ı Sergeev. On sets of eigenvalues of matrices with prescribed row sums and prescribed graph. Linear Algebra Appl., 455:187-209, 2014.

    [FM10] S´ebastien Ferenczi and Thierry Monteil. Infinite words with uniform frequencies, and invariant measures. In Combinatorics, automata and number theory, volume 135 of Encyclopedia Math. Appl., pages 373-409. Cambridge Univ. Press, Cambridge, 2010.

    [HY11] Masaki Hama and Hisatoshi Yuasa. Invariant measures for subshifts arising from substitutions of some primitive components. Hokkaido Math. J., 40(2):279-312, 2011.

    [Lem06] Bas Lemmens. Nonlinear Perron-Frobenius theory and dynamics of cone maps. In Positive systems, volume 341 of Lecture Notes in Control and Inform. Sci., pages 399-406. Springer, Berlin, 2006.

    [LU15] Martin Lustig and Caglar Uyanik. North-south dynamics of hyperbolic free group automorphisms on the space of currents. arXiv preprint arXiv:1509.05443, 2015.

    [Que10] Martine Queff´elec. Substitution dynamical systems-spectral analysis, volume 1294 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, second edition, 2010.

  • Related Organizations (3)
  • Metrics