A smooth function $f: \mathbb R^2 \to \mathbb R^2$ preserves the unit circle and satisfies $Df=2I$ on the unit circle. I want to decide whether the following is true:
Every two points $z,w$ on the unit circle have an iterate $f^{\circ n}$ for which $$|f^{\circ n}(z)-f^{\circ n}(w)| \geq \frac 1 {100}.$$
Intuitively if the points are too close then applying $f$ should double their distance, and we keep doubling until they are far apart. But I struggle with making this argument precise.