For the $\beta$-transformations themselves (including nonlinear versions with
the same coding space),
[Walters1978]
gives unique equilibrium
state for every Lipschitz potential; this was extended to Holder potentials in
[CT13]
. 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
[Buzzi2004]
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 $x\mapsto \alpha + \beta x \pmod 1$.
Comments
cf. [hofbauer1982ergodic] and question 111.
For the $\beta$-transformations themselves (including nonlinear versions with the same coding space), [Walters1978] gives unique equilibrium state for every Lipschitz potential; this was extended to Holder potentials in [CT13] . 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 [Buzzi2004] 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 $x\mapsto \alpha + \beta x \pmod 1$.