JournalsjfgVol. 1, No. 1pp. 83–152

Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets

  • Antti Käenmäki

    University of Jyväskylä, Finland
  • Henry W. J. Reeve

    University of Bristol, Great Britain
Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets cover
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Abstract

We develop a thermodynamic formalism for quasi-multiplicative potentials on a countable symbolic space and apply these results to the dimension theory of infinitely generated self-affine sets. The first application is a generalisation of Falconer's dimension formula to include typical infinitely generated self-affine sets and show the existence of an ergodic invariant measure of full dimension whenever the pressure function has a root. Considering the multifractal analysis of Birkhoff averages of general potentials Φ\Phi taking values in RN\mathbb{R}^{\mathbb{N}}, we give a formula for the Hausdorff dimension of JΦ(α)J_\Phi(\alpha), the α\alpha-level set of the Birkhoff average, on a typical infinitely generated self-affine set. We also show that for bounded potentials Φ\Phi, the Hausdorff dimension of JΦ(α)J_\Phi(\alpha) is given by the maximum of the critical value for the pressure and the supremum of Lyapunov dimensions of invariant measures μ\mu for which Φdμ=α\int\Phi\,d\mu=\alpha. Our multifractal results are new in both the finitely generated and the infinitely generated setting.

Cite this article

Antti Käenmäki, Henry W. J. Reeve, Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets. J. Fractal Geom. 1 (2014), no. 1, pp. 83–152

DOI 10.4171/JFG/3