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Rotation, entropy, and equilibrium states


Author: Oliver Jenkinson
Journal: Trans. Amer. Math. Soc. 353 (2001), 3713-3739
MSC (2000): Primary 54H20, 37C45, 28D20
DOI: https://doi.org/10.1090/S0002-9947-01-02706-4
Published electronically: April 18, 2001
MathSciNet review: 1837256
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Abstract: For a dynamical system $(X,T)$ and function $f:X\to\mathbb{R} ^d$ we consider the corresponding generalised rotation set. This is the convex subset of $\mathbb{R} ^d$ consisting of all integrals of $f$ with respect to $T$-invariant probability measures. We study the entropy $H(\varrho)$of rotation vectors $\varrho$, and relate this to the directional entropy $\mathcal{H}(\varrho)$ of Geller & Misiurewicz. For $(X,T)$ a mixing subshift of finite type, and $f$ of summable variation, we prove that if the rotation set is strictly convex then the functions $\mathcal{H}$ and $H$ are in fact one and the same. For those rotation sets which are not strictly convex we prove that $\mathcal{H}(\varrho)$ and $H(\varrho)$can differ only at non-exposed boundary points $\varrho$.


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Additional Information

Oliver Jenkinson
Affiliation: UPR 9016 CNRS, Institut de Mathématiques de Luminy, 163 avenue de Luminy, case 907, 13288 Marseille, cedex 9, France
Address at time of publication: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
Email: omj@maths.qmw.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-01-02706-4
Received by editor(s): November 22, 1999
Received by editor(s) in revised form: April 13, 2000
Published electronically: April 18, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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