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 and Continuous Dynamical Systems, 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: 1112.5968

[2]

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

[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.

[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.

[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.

[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.

[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.

[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, 2006

[9]

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

[10]

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

[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.

[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.

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: 1112.5968

[2]

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

[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.

[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.

[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.

[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.

[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.

[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, 2006

[9]

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

[10]

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

[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.

[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.

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