Equilibrium states of generalised singular value potentials and applications to affine iterated function systems

Jairo Bochi, Ian D. Morris

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)995-1028
Number of pages34
JournalGeometric and Functional Analysis
Issue number4
StatePublished - Jul 1 2018

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology

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