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A remark on the topological entropy and invariant circles of an area preserving twistmap. (English) Zbl 0769.58036
Twist mappings and their applications, Proc. Workshop, Minneapolis/MN (USA) 1989/90, IMA Vol. Math. Appl. 44, 1-5 (1992).
[For the entire collection see Zbl 0745.00054.]
Area preserving twist homeomorphisms \(f: A\longrightarrow A\) of the annulus \(A = S^ 1 \times {[0,1]}\) are considered such that the boundary components \(A_ j = S^ 1\times\{j\}\) are invariant under \(f\). For such an \(f\) let \(\rho_ j\) be the rotation number of the restriction of \(f\) on \(A_ j\). It is proved that if \(h_{top} (f) = 0\), then for every \(\omega\in (\rho_ 0,\rho_ 1)\), \(f\) has an invariant circle of rotation number \(\omega\). Two proofs of this fact are presented the first of which is a simple consequence of results of D. Hall and P. Boyland.
Reviewer: L.Stoyanov (Perth)

37A99 Ergodic theory
54C70 Entropy in general topology