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
In this paper we prove that translation structures for which the corresponding vertical translation flows is weakly mixing and disjoint with its inverse, form a G δ -dense set in every non-hyperelliptic connected component of the moduli space M. This is in contrast to hyperelliptic case, where for every translation structure the associated vertical flow is isomorphic to its inverse. To prove the main result, we study limits of the offdiagonal 3-joinings of special representations of vertical translation flows. Moreover, we construct a locally defined continuous embedding of the moduli space into the space of measure-preserving flows to obtain the G δ -condition.
We study self-similarity problem for two classes of flows:(1) special flows over circle rotations and under roof functions with symmetric logarithmic singularities (2) special flows over interval exchange transformations and under roof functions which are of two types• piecewise constant with one additional discontinuity which is not a discontinuity of the IET; • piecewise linear over exchanged intervals with non-zero slope.We show that if {T α,f t } t∈R is as in (1), then for a full measure set of rotations, and for every K, L ∈ N, K = L, we have that {T α,f Kt } t∈R and {T α,f Lt } t∈R are spectrally disjoint. Similarly, if {T f t } t∈R is as in (2), then for a full measure set of IET's, almost every position of the additional discontinuity (of f , in piecewise constant case) and every K, L ∈ N, K = L the flows {T f Kt } t∈R and {T f Lt } t∈R are spectrally disjoint.
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