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

Abstract

Let T = (T f t ) t∈R 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 so-called switchable Ratner's property which was introduced in [4]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [31] and not mixing [32]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows which are mildly mixing and not mixing.

Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows

“…Finally, by (16) and (19) it follows that |W (M)| |V (M)| + p + 2ǫ 2 2V + p and hence W (M) ∈ P . This finishes the proof.…”

“…Notice that, for any given N ∈ N, there exists δ N > 0 such that, for δ ∈ (0, δ N ) we have M N, and by Lemma 5.5 (for a 1,M , a 2,M ), by the definition of T > 0, there exists V > 0, independent of ǫ > 0, δ > 0 and N ∈ N, such that |a M | V, for some V > 0. This gives (16).…”

“…In what follows Φ : (X, B, µ, d) → (X, B, µ, d) is an ergodic automorphism of a σcompact metric probability space and f ∈ L + 1 (X). We have the following proposition (see Proposition 4.1. in [16]):…”

“…In [16] Proposition 3.2 was proved for the SR-property, which is a modification of Ratner's property in which one allows for divergence either in the future or in the past. However the proof in [16] immediately extends to a proof of Proposition 3.2.…”

Mild mixing of certain interval-exchange transformations

“…Using this criterion Frączek [Fra09] has shown that, when the genus is at least two, the set of Abelian differentials for which the vertical flow is mild mixing is dense in every stratum of moduli space. More recently, Kanigowski and Kułaga-Przymus [KKP15] showed that roof functions over interval exchange transformation having symmetric logarithmic singularities at some of the discontinuities of the interval exchange transformation give rise to special flows that are mild mixing.…”