Mild mixing property for special flows under piecewise constant functions

Spectral Theory of Dynamical Systems

“…In particular, R-property implies "rigidity" of joinings and it also implies the PID property; hence, mixing and R-property imply mixing of all orders. In Frączek and Lemańczyk (2006) and Frączek et al (2007), a version of R-property is shown for the class of von Neumann special flows (however, a is assumed to have bounded partial quotients). This allowed one to prove there that such flows are even mildly mixing (mixing is excluded by a Kochergin's result).…”

Slow chaos in surface flows

“…This changed drastically in the last decade. The first examples outside the homogeneous world were given by Frączek and Lemańczyk in [33][34][35] (in the setting of special flows). The two authors could also show in [33] that a variant of Ratner's property hold for some surface flows, more precisely in a class of flows on genus one tori known as von Neumann flows 28 (for non generic flows, corresponding to a measure zero set of frequencies).…”

“…6), one drastically looses control of the divergence. The Ratner property in its classical form (as well as the weaker versions defined in [33,34]) is expected to fail for of smooth area-preserving flows with non-degenerate fixed points. 29…”

“…Advances in our understanding of disjointness properties became possible building on the switchable Ratner property [6,19,29]. The notion of disjointness 34 was introduced in the 1970s by Furstenberg (see in particular [37]); two disjoint flows are in particular not isomorphic. 35 A disjointness property which has received a lot of attention recently (in particular as a tool in connection with Sarnak's conjecture on Moebius orthogonality, see below) is disjointness of rescalings.…”

“…The main result in [42] (which can be expressed in the language of special flows over IETs under roofs with asymmetric logarithmic singularities) on the other hand gives a full measure condition not only the higher genus case, but also the case of genus one and more saddles. 34 Two measure preserving flows ϕ R and φ R are called disjoint (in the sense of Fursterberg) if their only common ergodic joining is the trivial joining (i.e. the product joining).…”