On Bressan's conjecture on mixing properties of vector fields

In [9], the author considers a sequence of invertible maps Ti : S 1 → S 1 which exchange the positions of adjacent intervals on the unit circle, and defines as An the image of the set {0 ≤ x ≤ 1/2} under the action of Tn ◦ · · · ◦ T1 ,ş ťľ ł An = Tn ◦ · · · ◦ T1 x1 ≤ 1/2 .Then, if An is mixed up to scale h, it is proved that n ť Xş1(0.1) Tot.Var.(Ti − I) + Tot.Var.(T−1 − I) ≥ C log .ih i=1 We prove that (0.1) holds for general quasi incompressible invertible BV maps on R, and that this estimate implies that the map Tn ◦ · · · ◦ T1 belongs to the Besov space B 0,1,1 , and its norm is bounded by the sum of the total variation of T − I and T−1 − I, as in (0.1).