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Surface transformations and their dynamical applications

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Acta Mathematica

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Literatur

  1. Dynamical systems with two degrees of freedom. Transactions of the American Mathematical Society, vol. 18, 1917.

  2. Cf.E. Goursat,Sur les transformations ponctuelles qui conservent les volumes. Bulletin des Sciences Mathématiques, vol. 52, 1917.

  3. It should be observed that the definition refers to the vicinity of an invariant point.

  4. SeeT. Levi-Civita,Sopra alcuni criteri di instabilità. Annali di Matematica, Ser. III vol. 5, 1901.

  5. This fact has been noted byC. L. Bouton,Bulletin of the American Mathematical Society, vol. 23, 1916, p. 73. See alsoA. A. Bennett,A case of iteration in several variables, Annals of Mathematics, vol. 17, 1915–1916.

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  6. C. L. Bouton observed these facts in case II″,loc. cit..

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  7. Sur les courbes définies par les équations différentielles, Journal de mathématiques, ser. 3, vols. 7–8, 1881–1882 and ser. 4, vols. 1–2, 1885–1886. The analogy was explained partially by means of a limiting process byS. Lattés,Sur les équations fonctionelles qui définissent une courbe ou une surface invariante par une transformation, Annali di Matematica, ser. 3, vol. 13, 1907.

  8. This is the «symbol of the infinitesimal transformation» in the terminology ofLie.

  9. Cf.W. F. Osgood,Factorization of analytic functions of several variables, Annals of Mathematics, vol. 19, 1917–1918.

  10. The presence of fractional powers means that the root indicated is to be formally extracted.

  11. Sur l’itération et les solutions asymptotiques des équations différentielles, Bulletin de la Société Mathématique de France, vol. 29, 1901.

  12. The linear terms taken are clearly large enough. The coefficients ofu m v n (m+n ≧2) in either series is at least as large asKL m+n−2, which evidently exceeds numerically the coefficient ofu m v n inu 1 orv 1 ifK, L be chosen sufficiently large to begin with.

  13. For a simple development of the properties of ψ (z) used here, seeK. P. Williams,The asymptotic form of the function ψ (x). Bulletin of the American Mathematical Society, vol. 19, 1912–1913

  14. SeeJ. F. Ritt,On the differentiability of asymptotic series. Bulletin of the American Mathematical Society, vol. 24, 1917–1918, for a discussion of such differentiation.

  15. Cf.W. F. Osgood,On the uniformisation of algebraic functions, Annals of Mathematics, vol. 14, 1912–1913, pp. 152–154.

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  16. It is apparent from this result that no other invariant curves through the invariant point can exist.

  17. Levi-Civita (loc. cit.)Sopra alcuni criteri di instabilità. Annali di Matematica, Ser. III vol. 5, 1901. proved that certain nearly points are carried away from the invariant point in this and other hyperbolic cases, showing that the point is unstable. See alsoA. R. Cigala,Sopra un criterio di instabilità, Annali di Matematica, ser. 3, vol. 11, 1905.

  18. Compare the method of proof with a proof given byH. Poincaré,Les methodes nouvelles de la mécanique céleste, vol. 3, Paris, 1899, pp. 149–151.

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  19. Introduced byPoincaré (loc. cit.Les methodes nouvelles de la mécanique céleste, vol. 3, Paris, 1899, pp. 149–151. § 8).

  20. The assumptionc>0 is still made. This entails no specialization of course.

  21. That is, a chain ofa (or ω) points, each point arbitrarily near its successor, extending fromr=0 tor=d, can be found.

  22. See my paper first cited.

  23. See my paper first cited.

  24. Continuous one-one transformations of surfaces in themselves, Proceedings of the Section of Sciences, Koninklijke Academie van Wetenschappen te Amsterdam, vols. 11–15 (1908–1912). In the last part of this paperBrouwer develops the notion ofclass of a transformation, given later by myself in the paper first cited without knowledge of his work.

  25. SeeG. D. Birkhoff,Quelques théorèmes sur le mouvement des systèmes dynamiques, Bulletin de la Société Mathématique de France, vol. 40, 1912. The reader will observe the complete analogy between the recurrent motions of that paper and recurrent point groups.

  26. A periodic point group ofq pointsP, T(P),..., T q−1 (P) is called hyperbolic ifP is hyperbolic underT q . A similar terminology is employed in generab.

  27. The exceptional case in which there is a single recurrent point group whose minimal set fillsS is left out of consideration.

  28. See my paper first cited.

  29. In the simplest and general case I′,x, y may be expressed as convergent power series in ρ±τ while we havet=cτ+another power series of the same sort.

  30. H. Poincaré,Les méthodes nouvelles de la mécanique céleste, vol. 1, Paris 1892, Chap. 5.

  31. See my paper last cited.

  32. The existence of recurrent motions of discontinuous type has been established byH. C. M. Morse,Certain types of geodesic motion on a surface of negative curvature, Harvard Dissertation, 1917.

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    George D. Birkhoff

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Birkhoff, G.D. Surface transformations and their dynamical applications. Acta Math. 43, 1–119 (1922). https://doi.org/10.1007/BF02401754

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