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On Bressan's conjecture on mixing properties of vector fields

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Bianchini S.

Banach Center Publications

2006

tom 74

nr 1

13-31

In [9], the author considers a sequence of invertible maps $T_i : S¹ → S¹$ 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) $∑_{i=1}^{n} (Tot.Var.(T_i - I) + Tot.Var.(T_i^{-1} - I)) ≥ Clog 1/h$. 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 by the sum of the total variation of T - I and $T^{-1} - I$, as in (2).

Ograniczanie wyników

1 Banach Center Publications

1 Bianchini S.

1 2006

Wyniki wyszukiwania

On Bressan's conjecture on mixing properties of vector fields