The Pressure Function for Products of Non-negative Matrices

Equilibrium states of the pressure function for products of matrices

Let $\{M_i\}_{i=1}^\ell$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1... i_n\in \{1,..., \ell\}^n$ such that $M_{i_1}... M_{i_n}\neq {\bf 0}$. Let $P \colon (0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^\ell$. 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.

Remarks on the analyticity of subadditive pressure for products of triangular matrices

We study Falconer's subadditive pressure function with emphasis on analyticity. We begin by deriving a simple closed form expression for the pressure in the case of diagonal matrices and, by identifying phase transitions with zeros of Dirichlet polynomials, use this to deduce that the pressure is piecewise real analytic. We then specialise to the iterated function system setting and use a result of Falconer and Miao to extend our results to include the pressure for systems generated by matrices which are simultaneously triangularisable. Our closed form expression for the pressure simplifies a …

An inequality for the matrix pressure function and applications

We prove an a priori lower bound for the pressure, or $p$-norm joint spectral radius, of a measure on the set of $d \times d$ real matrices which parallels a result of J. Bochi for the joint spectral radius. We apply this lower bound to give new proofs of the continuity of the affinity dimension of a self-affine set and of the continuity of the singular-value pressure for invertible matrices, both of which had been previously established by D.-J. Feng and P. Shmerkin using multiplicative ergodic theory and the subadditive variational principle. Unlike the previous proof, our lower bound yields…

矩阵乘积压力函数的解析性

Analyticity of the pressure function for products of matrices

本文研究非可逆矩阵乘积的压力函数，在不可约与压缩假设下证明其解析性，并建立变分原理以推广已有结果。

Pressure at infinity and strong positive recurrence in negative curvature

In the context of geodesic flows of noncompact negatively curved manifolds, we propose three different definitions of entropy and pressure at infinity, through growth of periodic orbits, critical exponents of Poincaré series, and entropy (pressure) of invariant measures. We show that these notions coincide. Thanks to these entropy and pressure at infinity, we investigate thoroughly the notion of strong positive recurrence in this geometric context. A potential is said strongly positively recurrent when its pressure at infinity is strictly smaller than the full topological pressure. We show in …

随机矩阵乘积的Lyapunov指数

Lyapunov exponents for random products of non-negative matrices

本文研究了随机矩阵乘积的Lyapunov指数，并给出了Cantor集交集的Hausdorff维数公式及递推关系的增长速率。