In this paper we solve two open problems in ergodic theory. We prove first that if a Doeblin function g (a g-function) satisfies \(\begin{aligned} \limsup _{n\rightarrow \infty }\frac{\operatorname {var}_n \log g}{n^{-1/2}} < 2, \end{aligned}\) then we have a unique Doeblin measure (g-measure). This result indicates a possible phase transition in analogy with the long-range Ising model. Secondly, we provide an example of a Doeblin function with a unique Doeblin measure that is not weakly mixing, which implies that the sequence of iterates of the transfer operator does not converge, solving a well-known folklore problem in ergodic theory. Previously it was only known that uniqueness does not imply the Bernoulli property.
