JournalsaihpcVol. 34, No. 2pp. 407–421

Transience and multifractal analysis

  • Godofredo Iommi

    Facultad de Matemáticas, Pontificia Universidad Católica de Chile (PUC), Avenida Vicuña Mackenna, 4860 Santiago, Chile

  • Thomas Jordan

    The School of Mathematics, The University of Bristol, University Walk, Clifton, Bristol, BS8 1TW, UK

  • Mike Todd

    Mathematical Institute, University of St Andrews, North Haugh, St Andrews, KY16 9SS, Scotland, UK
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Abstract

We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.

Cite this article

Godofredo Iommi, Thomas Jordan, Mike Todd, Transience and multifractal analysis. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 2, pp. 407–421

DOI 10.1016/J.ANIHPC.2015.12.007