Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles

By De-Jun Feng, Pablo Shmerkin. 2013

58 citations recorded by the source corpus.

The pressure function P(A, s) plays a fundamental role in the calculation of the dimension of “typical” self-affine sets, where A = (A1, …,Ak) 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4ea66faaf50effb026ec63f5a28c42d3c492432e

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In Applied Mathematics

1. Ian D. Morris
2. Pablo Shmerkin
3. P. P. Varj'u

In Mathematical Physics

1. Ian D. Morris
2. Pablo Shmerkin
3. Meng Wu
4. P. P. Varj'u
5. Michael Hochman
6. Jonathan M. Fraser

Citations in the corpus

• On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems (2016, background): “, AM ) are in fact dense in large open subsets of parameter space: firstly, in the set of all tuples of affinity dimension strictly greater than 3/2 (which is open since affinity dimension is continuous, see [14]); and secondly, in the set of all tuples satisfying the bunching condition of Theorem 1”