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Proof of Poincaré's Geometric Theorem,Transactions of the American Mathematical Society, volume 14; or see a translation in volume 42 of theBulletin de la Société Mathématique de France.
Aclosed curve will be defined as the common boundary of a finite, simply connected, open continuum and the complementary open outer continuum. Aring is the region bounded by two closed curves, one within the other. If these curves do not touch, the ring is a doubly connected open continuum. No other type of ring enters here until the last section 8.
The restriction made on the curvesΓ andΓ 1 might be lightened in that these curves need only to be «right-handedly accessible» and «left-handedly accessible», as these terms are defined in my paper, «Surface Transformations and their Dynamical Applications» in volume 43 of theActa mathematica. But the less general and somewhat simpler theorem stated suffices to illustrate the same type of extension, and appears to be adequate for the dynamical applications.
The notable investigations ofH. Bohr have taken up the analytic representation of such motions. See, for instance, his recent papers: Zur Theorie der fast periodischer Funktionen, volume 45,Acta mathematica; Einige Sätze über Fourierreihe fastperiodischer Funktionen, volume23, Mathematische Zeitschrift.
In my Chicago Colloquium Lectures onDynamical Systems, soon to appear in book form, I establish these assertions.
The case where there are infinitely many invariant points may be excluded from consideration.
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Birkhoff, G.D. An extension of Poincaré's last geometric theorem. Acta Math. 47, 297–311 (1926). https://doi.org/10.1007/BF02559515
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DOI: https://doi.org/10.1007/BF02559515