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).

On Bressan's conjecture on mixing properties of vector fields / Bianchini, Stefano. - 74:(2006), pp. 13-31. [10.4064/bc74-0-1]

On Bressan's conjecture on mixing properties of vector fields

Bianchini, Stefano
2006-01-01

Abstract

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).
2006
74
Self-similar solutions of nonlinear PDE
13
31
Bianchini, Stefano
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/13100
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