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On Bressan's conjecture on mixing properties of vector fields

Stefano Bianchini (2006)

Banach Center Publications

In [9], the author considers a sequence of invertible maps T i : 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, (1) Aₙ = (Tₙ ∘ ... ∘ T₁)x₁ ≤ 1/2. Then, if Aₙ is mixed up to scale h, it is proved that (2) i = 1 n ( T o t . V a r . ( T i - I ) + T o t . V a r . ( T i - 1 - I ) ) C l o g 1 / h . We prove that (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...

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