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Compensation functions for factors of shifts of finite type
Published online by Cambridge University Press: 02 October 2014
Abstract
Let ${\it\pi}:X\rightarrow Y$ be an infinite-to-one factor map, where
$X$ is a shift of finite type. A compensation function relates equilibrium states on
$X$ to equilibrium states on
$Y$. The
$p$-Dini condition is given as a way of measuring the smoothness of a continuous function, with
$1$-Dini corresponding to functions with summable variation. Two types of compensation functions are defined in terms of this condition. Given a fully supported invariant measure
${\it\nu}$ on
$Y$, we show that the relative equilibrium states of a
$1$-Dini function
$f$ over
${\it\nu}$ are themselves fully supported, and have positive relative entropy. We then show that there exists a compensation function which is
$p$-Dini for all
$p>1$ which has relative equilibrium states supported by a subshift on which
${\it\pi}$ is a finite-to-one map onto
$Y$.
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- Research Article
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- © Cambridge University Press, 2014
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