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

Anna Florio

doi:10.3934/jmd.2022017

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2022, Volume 18: 555-573. Doi: 10.3934/jmd.2022017

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

Anna Florio

Université Paris Dauphine – Ceremade UMR 7534, Place du M^al De Lattre De Tassigny, 75775 PARIS Cedex 16, France

Received: January 17, 2020

Revised: July 18, 2022

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Abstract

For negative-torsion maps on the annulus we show that on every $ \mathscr{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 $ \mathscr{C}^1 $ essential curves in the framework of negative-torsion maps.

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Mathematics Subject Classification: Primary: 37E40; Secondary: 37C05.

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Figure 1. Phase portrait of the pendulum system of Example 2.6

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Figure 2. The simple curve built in the proof of Proposition 3.10

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