On the set of points of zero torsion for negative-torsion maps of the annulus

Abstract

For negative-torsion maps on the annulus we show that on every C 1 essential curve there is at least one point of zero torsion. As an outcome we deduce that the Hausdorff dimension of the set of points of zero torsion is greater or equal 1. As a byproduct we obtain a Birkhoff's-theorem-like result for C 1 essential curves in the framework of negative-torsion maps.

Torsion of instability zones for conservative twist maps on the annulus

“…the right inequality occurring because DI n (z,w) dθ < 0 if n m and DJ τ (z) (z, w) = DI mτ (z) (z, w) (see (7)) and the left inequality occurring because Torsion 1 ( f, z, w) > −1 for every (z, w) ∈ T 1 A. By Kac's lemma we know that τ is Leb-integrable and so that φ : (z, w) → DJ τ (z) (z,w) dθ is integrable and negative.…”

“…On the other hand, it can be proved that, for every n ∈ N * , there exists z ∈ A such that DI n (z,v) dθ ∈ (− 1 2 , 0), see [7]. More precisely, selecting a lift F of f, the linking number at any finite time n ∈ N * of the couple of points (x, r 0 ), (x + 1, r 0 ), for some x ∈ R, r 0 ∈ R, is zero.…”