2017/10/12 by Jairo Bochi, Bochi, Jairo, Ian D. Morris +1
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.DS
paper · pdf · doi:10.48550/arxiv.1710.04499
33 pages, 2 figures. V2: Minor changes. This is the final version, accepted for publication in GAFA
openalex publication_date 2017/10/12 · arxiv created 2018/03/21 · arxiv updated 2018/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We completely describe the equilibrium states of a class of potentials over the full shift which includes Falconer's singular value function for affine iterated function systems with invertible affinities. We show that the number of distinct ergodic equilibrium states of such a potential is bounded by a number depending only on the dimension, answering a question of A. Käenmäki. We prove that all such equilibrium states are fully supported and satisfy a Gibbs inequality with respect to a suitable subadditive potential. We apply these results to demonstrate that the affinity dimension of an iterated function system with invertible affinities is always strictly reduced when any one of the maps is removed, resolving a folklore open problem in the dimension theory of self-affine fractals. We deduce a natural criterion under which the Hausdorff dimension of the attractor has the same strict reduction property.