Transactions of the American Mathematical Society

Author(s): Oliver Jenkinson
Journal: Trans. Amer. Math. Soc. 353 (2001), 3713-3739.
MSC (2000): Primary 54H20, 37C45, 28D20
Posted: April 18, 2001
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Abstract: For a dynamical system

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

with respect to

-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

a mixing subshift of finite type, and

of summable variation, we prove that if the rotation set is strictly convex then the functions $\mathcal{H}$ and

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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