Continuity of sub-additive topological pressure with matrix cocycles * *The first author is partially supported by NSFC (11901305), the second author is partially supported by NSFC (11771317, 11790274), the third author is partially supported by NSFC (11871361, 11790274) and Qinglan project from Jiangsu Province.

Abstract

Let A = {A 1, A 2, …, A k } be a finite collection of contracting affine maps, the corresponding pressure function P(A, s) plays the fundamental role in the study of dimension of self-affine sets. The zero of the pressure function always give the upper bound of the dimension of a self-affine set, and is exactly the dimension of ‘typical’ self-affine sets. In this paper, we consider an expanding base dynamical system, and establish the continuity of the pressure with the singular value function of a Hölder continuous matrix cocycle. This extends Feng and Shmerkin’s result in (Feng and Shmerkin 2014 Geom. Funct. Anal. 24 1101–1128) to a general setting.