On special flows over IETs that are not isomorphic to their inverses

Abstract

In this paper we give a criterion for a special flow to be not isomorphic to its inverse which is a refine of a result in \cite{Fr-Ku-Le}. We apply this criterion to special flows $T^f$ built over ergodic interval exchange transformations $T:[0,1)\to[0,1)$ (IETs) and under piecewise absolutely continuous roof functions $f:[0,1)\to\mathbb{R}_+$. We show that for almost every IET $T$ if $f$ is absolutely continuous over exchanged intervals and has non-zero sum of jumps then the special flow $T^f$ is not isomorphic to its inverse. The same conclusion is valid for a typical piecewise constant roof function

On Typicality of Translation Flows Which Are Disjoint With Their Inverse

“…Therefore, by (27), (31) and Proposition 3.2, the flows T and T −1 = S are disjoint. The above criterion strengthens the results obtained in [3], that is the flows described in [3] are not only non-isomorphic with their inverses, but also disjoint. More precisely, one can prove the following results.…”

Spectral disjointness of rescalings of some surface flows

“…Finally, we also have the following result which is proven in [3]. Lemma Let be an IET such that is well defined.…”

“…What follows is the proof that the sums of heights of two dominating towers in the aforementioned choice of towers form a rigidity sequence of IET. This is also in contrast with [3] where authors obtained and used only sequence of partial rigidity. Furthermore, the set of proper piecewise constant roof functions is picked or rather the set of appropriate discontinuity points.…”

“…The following result was proven in [10] and in more general version in [3] and gives the description of the aforementioned limits under some natural assumptions. Recall that for a dynamical system (X, B, μ, T ), we say that {q n } n∈N is a rigidity sequence along a sequence of subsets {W n } n∈N if for every measurable A ⊂ B, we have…”

“…We would also like to refer the reader to the papers [9, 13] and [18] where the authors describe the set of self‐similarities for various classes of flows. Some of the results in this paper base on the constructions given in [3] where authors were focusing on relation of some special flows with their inverses, that is rescaling by .…”