Statistics plus dynamics of transformations of $[0,1]$ - 'non-linear' $\beta$-expansions like examples.
cf.  and question 111.
For the $\beta$-transformations themselves (including nonlinear versions with the same coding space),  gives unique equilibrium state for every Lipschitz potential; this was extended to Holder potentials in . 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  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 $x\mapsto \alpha + \beta x \pmod 1$.