The Pressure Function for Products of Non-negative Matrices

The Pressure Function for Products of Non-negative Matrices

Let (ΣA, σ) be a subshift of finite type and let M(x) be a continuous function on ΣA 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 M1, · · · ,Mm. The pres...

Contractive inhomogeneous products of non - negative matrices

Equilibrium States of the Pressure Function for Products of Matrices

Let {Mi}i=1 be a non-trivial family of d× d complex matrices, in the sense that for any n ∈ N, there exists i1 · · · in ∈ {1, . . . , `}n such that Mi1 · · ·Min 6= 0. Let P : (0,∞) → R be the pressure function of {Mi}i=1. 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.

Improvements on inequalities for non-negative matrices

We prove that there is an integer k :::; (n 2n + 4) /2 such that the diagonal entries of Ak are all positive for any non-negative irreducible n x n matrix A, and that there are integers i, j with 0 :::; i < j :::; 3n / 2 such that Ai :::; Aj for any non-negative n x n matrix A with no entry in (0,1) and n 2:: 2. The results of Wang and Shallit [Linear Algebra Appl. 290 (1999) 135-144] are thus ...

On Non-negative-valued Matrices.

cording to our Theorem, if (4) is fulfilled, f3 and a can be so chosen that (2) be fulfilled. This form shows, however, that the lowest term in the determinant Ioua'+ Owa-11, if considered as a polynomial in E, is {nT Hence, the lowest term in the determinant :u ± EWI is I0K-1Iaean-r, i.e., the coefficient of (n7r in the determinant |u + Ew| does not vanish. Conversely, let us assume that u and...