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 & Continuous Dynamical Systems - A, 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).   Google Scholar

[2]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture notes in Math., 470 (1975).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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, (2004), 181.   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

E. Le Page, "Théorèmes Limites pour les Produits de Matrices Aléatoires,", Lecture Notes in Math., 928 (1982).   Google Scholar

[18]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", in, 5 (1978).   Google Scholar

[19]

P. Walters, "An Introduction to Ergodic Theory,'', Springer-Verlag, (1982).   Google Scholar

show all references

References:
[1]

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

[2]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture notes in Math., 470 (1975).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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, (2004), 181.   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

E. Le Page, "Théorèmes Limites pour les Produits de Matrices Aléatoires,", Lecture Notes in Math., 928 (1982).   Google Scholar

[18]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", in, 5 (1978).   Google Scholar

[19]

P. Walters, "An Introduction to Ergodic Theory,'', Springer-Verlag, (1982).   Google Scholar

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