August  2011, 30(3): 699-708. doi: 10.3934/dcds.2011.30.699

Equilibrium states of the pressure function for products of matrices

1. 

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

2. 

Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014, University of Jyväskylä, Finland

Received  April 2010 Revised  October 2010 Published  March 2011

Let $\{M_i\}_{i=1}^l$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1\cdots i_n\in \{1,\ldots, l\}^n$ such that $M_{i_1}\cdots M_{i_n}\ne $0. Let P : $(0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^l$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property.
Citation: De-Jun Feng, Antti Käenmäki. Equilibrium states of the pressure function for products of matrices. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 699-708. doi: 10.3934/dcds.2011.30.699
References:
[1]

P. Bougerol and J. Lacroix, "Products of Random Matrices with Applications to Schrödinger Operators," Birkhäuser, 1985.

[2]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture notes in Math., 470, Springer-Verlag, 1975.

[3]

Y. L. Cao, D. J. Feng and W. Huang, The thermodynamical formalism for submultiplicative potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657.

[4]

K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc., 103 (1988), 339-350. doi: 10.1017/S0305004100064926.

[5]

K. Falconer and A. Sloan, Continuity of subadditive pressure for self-affine sets, Real Analysis Exchange, 34 (2009), 413-427.

[6]

D. J. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices, Israel J. Math., 138 (2003), 353-376. doi: 10.1007/BF02783432.

[7]

D. J. Feng, The variational principle for products of non-negative matrices, Nonlinearity, 17 (2004), 447-457. doi: 10.1088/0951-7715/17/2/004.

[8]

D. J. Feng, Lyapunov exponents for products of matrices and multifractal analysis, part II: General matrices, Israel J. Math., 170 (2009), 355-394. doi: 10.1007/s11856-009-0033-x.

[9]

D. J. Feng, Equilibrium states for factor maps between subshifts, Adv. Math., 226 (2011), 2470-2502. doi: i:10.1016/j.aim.2010.09.012.

[10]

D. J. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials, Comm. Math. Phys., 297 (2010), 1-43. doi: 10.1007/s00220-010-1031-x.

[11]

D. J. Feng and K. S. Lau, The pressure function for products of non-negative matrices, Math. Res. Lett., 9 (2002), 363-378.

[12]

H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-468. doi: 10.1214/aoms/1177705909.

[13]

Y. Guivarc'h and E. Le Page, Simplicité de spectres de Lyapounov et propriété d'isolation spectrale pour une famille d'opérateurs de transfert sur l'espace projectif, in "Random Walks and Geometry," Walter de Gruyter GmbH & Co. KG, Berlin, (2004), 181-259.

[14]

Y. Heurteaux, Estimations de la dimension inférieure et de la dimension supérieure des mesures, Ann. Inst. Henri Poincaré, 34 (1998), 309-338. doi: 10.1016/S0246-0203(98)80014-9.

[15]

A. Käenmäki, On natural invariant measures on generalised iterated function systems, Ann. Acad. Sci. Fenn. Math., 29 (2004), 419-458.

[16]

A. Käenmäki and M. Vilppolainen, Dimension and measures on sub-self-affine sets, Monatsh. Math., 161 (2010), 271-293. doi: 10.1007/s00605-009-0144-9.

[17]

E. Le Page, "Théorèmes Limites pour les Produits de Matrices Aléatoires," Lecture Notes in Math., 928, Springer, Berlin-New York, 1982.

[18]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," in "Encyclopedia of Mathematics and its Applications," 5, Addison-Wesley Publishing Co., Reading, Mass., 1978.

[19]

P. Walters, "An Introduction to Ergodic Theory,'' Springer-Verlag, 1982.

show all references

References:
[1]

P. Bougerol and J. Lacroix, "Products of Random Matrices with Applications to Schrödinger Operators," Birkhäuser, 1985.

[2]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture notes in Math., 470, Springer-Verlag, 1975.

[3]

Y. L. Cao, D. J. Feng and W. Huang, The thermodynamical formalism for submultiplicative potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657.

[4]

K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc., 103 (1988), 339-350. doi: 10.1017/S0305004100064926.

[5]

K. Falconer and A. Sloan, Continuity of subadditive pressure for self-affine sets, Real Analysis Exchange, 34 (2009), 413-427.

[6]

D. J. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices, Israel J. Math., 138 (2003), 353-376. doi: 10.1007/BF02783432.

[7]

D. J. Feng, The variational principle for products of non-negative matrices, Nonlinearity, 17 (2004), 447-457. doi: 10.1088/0951-7715/17/2/004.

[8]

D. J. Feng, Lyapunov exponents for products of matrices and multifractal analysis, part II: General matrices, Israel J. Math., 170 (2009), 355-394. doi: 10.1007/s11856-009-0033-x.

[9]

D. J. Feng, Equilibrium states for factor maps between subshifts, Adv. Math., 226 (2011), 2470-2502. doi: i:10.1016/j.aim.2010.09.012.

[10]

D. J. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials, Comm. Math. Phys., 297 (2010), 1-43. doi: 10.1007/s00220-010-1031-x.

[11]

D. J. Feng and K. S. Lau, The pressure function for products of non-negative matrices, Math. Res. Lett., 9 (2002), 363-378.

[12]

H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-468. doi: 10.1214/aoms/1177705909.

[13]

Y. Guivarc'h and E. Le Page, Simplicité de spectres de Lyapounov et propriété d'isolation spectrale pour une famille d'opérateurs de transfert sur l'espace projectif, in "Random Walks and Geometry," Walter de Gruyter GmbH & Co. KG, Berlin, (2004), 181-259.

[14]

Y. Heurteaux, Estimations de la dimension inférieure et de la dimension supérieure des mesures, Ann. Inst. Henri Poincaré, 34 (1998), 309-338. doi: 10.1016/S0246-0203(98)80014-9.

[15]

A. Käenmäki, On natural invariant measures on generalised iterated function systems, Ann. Acad. Sci. Fenn. Math., 29 (2004), 419-458.

[16]

A. Käenmäki and M. Vilppolainen, Dimension and measures on sub-self-affine sets, Monatsh. Math., 161 (2010), 271-293. doi: 10.1007/s00605-009-0144-9.

[17]

E. Le Page, "Théorèmes Limites pour les Produits de Matrices Aléatoires," Lecture Notes in Math., 928, Springer, Berlin-New York, 1982.

[18]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," in "Encyclopedia of Mathematics and its Applications," 5, Addison-Wesley Publishing Co., Reading, Mass., 1978.

[19]

P. Walters, "An Introduction to Ergodic Theory,'' Springer-Verlag, 1982.

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