PermalinkSubmitted by Vaughn Climenhaga on Sat, 07/15/2017 - 10:55

For the $\beta$-transformations themselves (including nonlinear versions with the same coding space), [1] gives unique equilibrium state for every Lipschitz potential; this was extended to Holder potentials in [2]. Of course general piecewise expanding interval maps have a huge literature: in the transitive case, every Holder potential has a unique equilibrium state provided it satisfies the condition $\sup_\mu \int \phi\,d\mu < P(\phi)$, where the supremum is over all invariant measures. Buzzi conjectured in [3] that this holds for all Holder potentials, and proved it when the map is continuous (not just piecewise continuous). So far as I know the conjecture is still open even for the examples $x\mapsto \alpha + \beta x \pmod 1$.

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## FL

cf. [1] and question 111.

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## For the $\beta$

For the $\beta$-transformations themselves (including nonlinear versions with the same coding space), [1] gives unique equilibrium state for every Lipschitz potential; this was extended to Holder potentials in [2]. Of course general piecewise expanding interval maps have a huge literature: in the transitive case, every Holder potential has a unique equilibrium state provided it satisfies the condition $\sup_\mu \int \phi\,d\mu < P(\phi)$, where the supremum is over all invariant measures. Buzzi conjectured in [3] that this holds for all Holder potentials, and proved it when the map is continuous (not just piecewise continuous). So far as I know the conjecture is still open even for the examples $x\mapsto \alpha + \beta x \pmod 1$.

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