Rotation, entropy, and equilibrium states

Jenkinson, Oliver

doi:10.1090/S0002-9947-01-02706-4

Rotation, entropy, and equilibrium states
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Trans. Amer. Math. Soc. 353 (2001), 3713-3739 Request permission

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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