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Let ${\mathcal{T}}=(T_{t}^{f})_{t\in \mathbb{R}}$ be a special flow built over an IET $T:\mathbb{T}\rightarrow \mathbb{T}$ of bounded type, under a roof function $f$ with symmetric logarithmic singularities at a subset of discontinuities of $T$ . We show that ${\mathcal{T}}$ satisfies the so-called switchable Ratner’s property which was introduced in Fayad and Kanigowski [On multiple mixing for a class of conservative surface flows. Invent. Math. to appear]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations.J. Mod. Dynam. 3 (2009), 35–49] and not mixing [Ulcigrai. Absence of mixing in area-preserving flows on surfaces. Ann. of Math. (2) 173 (2011), 1743–1778]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows that are mildly mixing and not mixing.

Ratner’s property and mild mixing for smooth flows on surfaces