Analyticity of the pressure function for products of matrices

TL;DR

This paper proves the analyticity of the pressure function for products of non-invertible matrices under certain conditions and establishes a variational principle, advancing understanding in mathematical physics and fractal geometry.

Contribution

It introduces new conditions under which the pressure function is analytic and generalizes the variational principle for such functions.

Findings

Proved analyticity of the pressure function for specific matrix products.

Established a variational principle for the pressure function.

Extended previous results to non-invertible matrices with irreducibility and contractivity.

Abstract

The pressure function is a fundamental object in various areas of mathematics. Its regularity is studied to derive insights into phase transitions in certain physical systems or to determine the Hausdorff dimension of self-affine sets. In this paper, we prove the analyticity of the pressure function for products of non-invertible matrices satisfying an irreducibility and a contractivity assumptions. Additionally, we establish a variational principle for the pressure function, thereby generalizing previous results.