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Sub-actions for young towers

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  • Let $T:X\to X$ be a dynamical system, and $\phi: X\to \mathbb{R}$ a function on $X$. A function $\theta:X\to \mathbb{R}$ is called a sub-action if $\theta$ satisfies the equation

    $\phi \leq \theta \circ T - \theta + m(\phi, T)$

    where $m(\phi, T)=$sup{$\int \phi d\mu:\mu$ is an invariant probability measure for $ T$}. The existence and regularity of sub-actions are important for the study of optimizing measures. We prove the existence of Hölder sub-actions for Lipschitz functions on certain classes of Manneville-Pomeau type maps. We also construct locally Hölder sub-actions for Lipschitz functions on Young Towers. In some settings (uniform hyperbolicity and Manneville-Pomeau maps) this implies Hölder sub-actions for the underlying system modeled by the Tower.

    Mathematics Subject Classification: 37D20, 37D25.


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