The pressure function for products of non-negative matrices

By De-Jun Feng, Ka-Sing Lau. 2002

98 citations recorded by the source corpus.

Let $(\Sigma_A, \sigma)$ be a subshift of finite type and let $M(x)$ be a continuous function on $\Sigma_A$ taking values in the set of non-negative matrices. We extend the classical scalar pressure function to this new setting and prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where $M(x)$ takes finite values $M_1, ..., M_m$. The pressure function reduces to $P(q):=\lim_{n\to \infty}\frac{1}{n} \log \sum_{J \in \sum_{A, n}} \|M_J\|^q$. The expression is important when we consider the multifractal formalism for certain iterated function systems with overlaps.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/075d77c9646dd7c5572b12a96a6c69c37f43de95

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In Condensed Matter Physics

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Citations in the corpus

Equilibrium states of the pressure function for products of matrices (2010, methodology): “2 was first proved in [11] for non-negative matrices under a different irreducibility assumption (that is, there exists r ∈ N so that ∑r i=1(M1 + · · ·+ Ml) r is a strictly positive matrix).”
Lyapunov Spectrum of Asymptotically Sub-additive Potentials (2009, methodology): “This is mainly motivated by the recent works on the Lyapunov exponents of matrix products [26, 23, 24] and the Lyapunov exponents of differential maps on nonconformal repellers [6].”
The thermodynamic formalism for sub-additive potentials (2007, background): “For the above purpose, a thermodynamic formalism (pressure functions, Gibbs measures and the distribution of Lyapunov exponents) for non-negative matrix-valued potentials was developed in [13, 16 ].”
Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures (2006, background): “This example is related to work in [7] (see also [6]).”
Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles (2013, background): “The function M(A, ·) is differentiable whenever there is just one equilibrium measure, and this is known to hold under several conditions, such as irreducibility or strict positivity, see [23, 22].”
Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices (2003, background): “The present paper is a continuation of our work in [11, 14] for studying the upper Lyapunov exponents for products of matrices.”
Smoothness of the Lq‐Spectrum of Self‐Similar Measures with Overlaps (2003, result): “The following result (for general non-negative matrices) was proved by Feng and Lau [5].”
The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers (2005, methodology): “For all the Pisot reciprocals besides simple Pisot reciprocals, by extending an idea in this paper and using a result on the product of non-negative matrices in[15], Feng [14] recently proved that (q) is always differentiable forq > 0 and (1.”