Action and periodic orbits on annulus

2021/06/11 by Deng, Yanxia, Xia, Zhihong
#Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2106.06105

Abstract

We consider the classical problem of area-preserving maps on annulus \mathbbA = S¹ × [0, 1] . Let M_f be the set of all invariant probability measures of an area-preserving, orientation preserving diffeomorphism f on \mathbbA. Given any μ₁ and μ₂ in M_f, Franks \citeFranks1988\citeFranks1992, generalizing Poincaré's last geometric theorem (Birkhoff \citeBirkhoff1913), showed that if their rotation numbers are different, then f has infinitely many periodic orbits. In this paper, we show that if μ₁ and μ₂ have different actions, even if they have the same rotation number, then f has infinitely many periodic orbits. In particular, if the action difference is larger than one, then f has at least two fixed points. The same result is also true for area-preserving diffeomorphisms on unit disk, where no rotation number is available.

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