Non-conformal repellers and the continuity of pressure for matrix cocycles

Abstract

The pressure function $P(A,s)$ plays a fundamental role in the calculation of the dimension of "typical" self-affine sets, where $A=(A_1,\dots ,A_k)$ is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on $A$. As a consequence, we show that the dimension of "typical" self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles.

24 pages. v2: minor improvements, incorporates referee suggestions

References in corpus (4)

• Lyapunov spectrum of asymptotically sub-additive potentials
• Equilibrium states of the pressure function for products of matrices
• Dimension and measures on sub-self-affine sets
• Generalised dimensions of measures on almost self-affine sets

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• Genericity of dimension drop on self-affine sets
• Random affine code tree fractals: Hausdorff and affinity dimensions and pressure
• Analyticity of the affinity dimension for planar iterated function systems with matrices which preserve a cone
• Non-invertible planar self-affine sets
• Totally ergodic generalised matrix equilibrium states have the Bernoulli property
• $L^q$-spectra of box-like graph-directed self-affine measures: closed forms, with rotation