| The history of the problem comes from the work of V.Arnold where he
notices that some ergodic flows on surfaces can be represented as special
flows over the interval exchange maps for functions with symmetric or
asymmetric logarithmic singularities and asks a question about mixing
properties of such flows.
The first result belongs to A.Kocergin who proved the absence of mixing
for symmetric singularities and almost all circle rotations. The next
result in this issue belongs to Sinai and Khanin who proved mixing for
asymmetric singularities and almost all circle rotations. Recently
C.Ulcigrai proved mixing for almost all interval exchanges and asymmetric
singularities.
We prove the absence of mixing for the case of symmetric singularities
for almost all interval exshange of 4 intervals of the permutation type (4321).
The proof includes elements of graphs combinatorics, substitution systems,
and Kontsevich-Zorich induction. |
