A new topology is introduced on the group of $\mu$-preserving automorphisms of a Lebesgue space $(X, \Sigma, \mu)$ so that $f_k \rightarrow f$ if $\mu(f^n_kA \cap B) \rightarrow \mu(f^nA \cap B)$ uniformly in $n$ for all $A, B$ in $\Sigma$. The subspace of mixing automorphisms is a Baire space in the relative topology. A conjecture (about the extent to which a mixing stationary process is determined by its two-dimension distributions) is stated, which is true implies that the two-fold mixing automorphisms are of first category in the mixing ones. So if the conjecture is true then by Baire's Theorem there is a mixing but not two-fold mixing automorphism.
"Conjecture: In General a Mixing Transformation is Not Two-Fold Mixing." Ann. Probab. 13 (1) 310 - 313, February, 1985. https://doi.org/10.1214/aop/1176993084