On Bressan's conjecture on mixing properties of vector fields
Volume 74 / 2006
Abstract
In [9], the author considers a sequence of invertible maps $\mathbf T_i : S^1 \to S^1$ which exchange the positions of adjacent intervals on the unit circle, and defines as $A_n$ the image of the set $\{0 \leq x \leq 1/2\}$ under the action of $\mathbf T_n \circ \dots \circ \mathbf T_1$, $$ A_n = ( \mathbf T_n \circ \dots \circ \mathbf T_1 ) \{ x_1 \leq 1/2 \}.\tag1$$ Then, if $A_n$ is mixed up to scale $h$, it is proved that $$\sum_{i=1}^n ( \hbox{Tot.Var.}(\mathbf T_i-{\bf I}) + \hbox{Tot.Var.}(\mathbf T_i^{-1}-{\bf I}) ) \geq C \log \frac{1}{h}.\tag2$$ We prove that (1) holds for general quasi incompressible invertible BV maps on $\mathbb R$, and that this estimate implies that the map $\mathbf T_n \circ \dots \circ \mathbf T_1$ belongs to the Besov space $B^{0,1,1}$, and its norm is bounded by the sum of the total variation of $\mathbf T-{\bf I}$ and $\mathbf T^{-1}-{\bf I}$, as in (2).
