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

Let T =(T^f_t)_tϵℝ be a special flow built over an IET T :T→T of bounded type, under a roof function f with symmetric logarithmic singularities at a subset of discontinuities of T . We show that 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 areapreserving 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.