On Bressan's conjecture on mixing properties of vector fields

Stefano Bianchini

Banach Center Publications, Tome 72 (2006), p. 13-31 / Harvested from The Polish Digital Mathematics Library

Résumé

In [9], the author considers a sequence of invertible maps $T_{i} : S ¹ \to S ¹$ 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) $\sum_{i = 1}^{n} (T o t . V a r . (T_{i} - I) + T o t . V a r . (T_{i}^{- 1} - I)) \geq C l o g 1 / h$ . 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 by the sum of the total variation of T - I and $T^{- 1} - I$ , as in (2).

Publié le : 2006-01-01

Zbl 1108.35028

EUDML-ID : urn:eudml:doc:282120

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     author = {Stefano Bianchini},
     title = {On Bressan's conjecture on mixing properties of vector fields},
     journal = {Banach Center Publications},
     volume = {72},
     year = {2006},
     pages = {13-31},
     zbl = {1108.35028},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc74-0-1}
}

Stefano Bianchini. On Bressan's conjecture on mixing properties of vector fields. Banach Center Publications, Tome 72 (2006) pp. 13-31. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc74-0-1/