January  2017, 37(1): 355-385. doi: 10.3934/dcds.2017015

Perron-Frobenius theory and frequency convergence for reducible substitutions

1. 

Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France

2. 

Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA

* Corresponding author: Caglar Uyanik

Received  June 2016 Revised  September 2016 Published  November 2016

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 and hence invariant measures on the associated subshift. The analogous results are well known for primitive substitutions and have found many applications, but for reducible substitutions the tools provided here were so far missing from the theory.

Citation: Martin Lustig, Caglar Uyanik. Perron-Frobenius theory and frequency convergence for reducible substitutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 355-385. doi: 10.3934/dcds.2017015
References:
[1]

M. Akian, S. Gaubert and R. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, arXiv: 1112.5968Google Scholar

[2]

N. Bédaride, A. Hilion and M. Lustig, Invariant measures for train track towers, arXiv: 1503.08000Google Scholar

[3]

S. BezuglyiJ. KwiatkowskiK. Medynets and B. Solomyak, Invariant measures on stationary Bratteli diagrams, Ergodic Theory Dynam. Systems, 30 (2010), 973-1007. doi: 10.1017/S0143385709000443. Google Scholar

[4]

P. ButkovičH. Schneider and S. Sergeev, Z-matrix equations in max-algebra, nonnegative linear algebra and other semirings, Linear Multilinear Algebra, 60 (2012), 1191-1210. doi: 10.1080/03081087.2012.656107. Google Scholar

[5]

G. M. EngelH. Schneider and S. Sergeev, On sets of eigenvalues of matrices with prescribed row sums and prescribed graph, Linear Algebra Appl., 455 (2014), 187-209. doi: 10.1016/j.laa.2014.05.010. Google Scholar

[6]

S. Ferenczi and T. 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.Google Scholar

[7]

M. Hama and H. Yuasa, Invariant measures for subshifts arising from substitutions of some primitive components, Hokkaido Math. J., 40 (2011), 279-312. doi: 10.14492/hokmj/1310042832. Google Scholar

[8]

B. 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, 2006Google Scholar

[9]

M. Lustig and C. Uyanik, North-south dynamics of hyperbolic free group automorphisms on the space of currents, arXiv: 1509.05443Google Scholar

[10]

M. Queffélec, Substitution Dynamical Systems—Spectral Analysis, volume 1294 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, second edition, 2010.Google Scholar

[11]

U. G. Rothblum, Algebraic eigenspaces of nonnegative matrices, Linear Algebra and Appl., 12 (1975), 281-292. doi: 10.1016/0024-3795(75)90050-6. Google Scholar

[12]

H. Schneider, The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: A survey, Proceedings of the symposium on operator theory (Athens, 1985), 84 (1986), 161-189. doi: 10.1016/0024-3795(86)90313-7. Google Scholar

show all references

References:
[1]

M. Akian, S. Gaubert and R. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, arXiv: 1112.5968Google Scholar

[2]

N. Bédaride, A. Hilion and M. Lustig, Invariant measures for train track towers, arXiv: 1503.08000Google Scholar

[3]

S. BezuglyiJ. KwiatkowskiK. Medynets and B. Solomyak, Invariant measures on stationary Bratteli diagrams, Ergodic Theory Dynam. Systems, 30 (2010), 973-1007. doi: 10.1017/S0143385709000443. Google Scholar

[4]

P. ButkovičH. Schneider and S. Sergeev, Z-matrix equations in max-algebra, nonnegative linear algebra and other semirings, Linear Multilinear Algebra, 60 (2012), 1191-1210. doi: 10.1080/03081087.2012.656107. Google Scholar

[5]

G. M. EngelH. Schneider and S. Sergeev, On sets of eigenvalues of matrices with prescribed row sums and prescribed graph, Linear Algebra Appl., 455 (2014), 187-209. doi: 10.1016/j.laa.2014.05.010. Google Scholar

[6]

S. Ferenczi and T. 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.Google Scholar

[7]

M. Hama and H. Yuasa, Invariant measures for subshifts arising from substitutions of some primitive components, Hokkaido Math. J., 40 (2011), 279-312. doi: 10.14492/hokmj/1310042832. Google Scholar

[8]

B. 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, 2006Google Scholar

[9]

M. Lustig and C. Uyanik, North-south dynamics of hyperbolic free group automorphisms on the space of currents, arXiv: 1509.05443Google Scholar

[10]

M. Queffélec, Substitution Dynamical Systems—Spectral Analysis, volume 1294 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, second edition, 2010.Google Scholar

[11]

U. G. Rothblum, Algebraic eigenspaces of nonnegative matrices, Linear Algebra and Appl., 12 (1975), 281-292. doi: 10.1016/0024-3795(75)90050-6. Google Scholar

[12]

H. Schneider, The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: A survey, Proceedings of the symposium on operator theory (Athens, 1985), 84 (1986), 161-189. doi: 10.1016/0024-3795(86)90313-7. Google Scholar

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