On the Vertical Rotation Intervals of Torus Homeomorphisms Homotopic to Dehn Twists

Abstract

In this paper, the vertical rotation interval of a homeomorphism $f:\mathbb {T}^2\rightarrow \mathbb {T}^2$ homotopic to Dehn twist is considered. We first prove that if the vertical rotation interval of f has an irrational endpoint, then we can find a homeomorphism g arbitrarily $C^0$-close to f such that $\rho _V(G)\cap \rho _V(F)^c\ne \varnothing $. It is also proved that the vertical displacement of points under n iterates of a lift of an area-preserving $C^{1+\epsilon }$-diffeomorphism f has bounded drift away from $n\rho $ at any $\rho \in \partial \rho _V(F)$ when $\rho _V(F)$ has interior. We also investigate the vertical rotation sets of minimal sets, which are proved to be closed intervals (possibly non-degenerate). Further, we show that if f has bounded mean vertical-motion on a minimal set $\mathcal {M}$, then $f|_{\mathcal {M}}$ is regularly semi-conjugate to the irrational rotation on $\mathbb {T}^1$.

Keywords

Dehn twist / Vertical rotation interval / Stability / 37A35 / 37E45

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Funding

National Natural Science Foundation of China(12471187)

Natural Science Foundation of Fujian Province(2023J05009)

Presidential Research Foundation of Xiamen University(20720240134)

Rights & permissions

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature