An inequality for the matrix pressure function and applications

TL;DR

This paper establishes a new lower bound for the matrix pressure function, enabling rigorous computation and providing insights into the continuity and discontinuities of related spectral functions for matrix sets.

Contribution

It introduces a novel a priori lower bound for the matrix pressure, leading to algorithms for precise computation and new proofs of continuity properties.

Findings

New lower bound for matrix pressure and singular value pressure.

Algorithms for computing pressure and affinity dimension with guaranteed accuracy.

Characterization of discontinuities in the singular value pressure function.

Abstract

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 algorithms to rigorously compute the pressure, singular value pressure and affinity dimension of a finite set of matrices to within an a priori prescribed accuracy in finitely many computational steps. We additionally deduce a related inequality for the singular value pressure for measures on the set of $2 \times 2$ real matrices, give a precise characterisation of the discontinuities of the singular value pressure function for two-dimensional matrices, and prove a general theorem relating the zero-temperature limit of the matrix pressure to the joint spectral radius.