Poincaré-Birkhoff Theorems in random dynamics

Pelayo, Álvaro; Rezakhanlou, Fraydoun

doi:10.1090/tran/6967

Poincaré-Birkhoff Theorems in random dynamics
HTML articles powered by AMS MathViewer

by Álvaro Pelayo and Fraydoun Rezakhanlou PDF

Trans. Amer. Math. Soc. 370 (2018), 601-639 Request permission

Abstract:

We propose a generalization of the Poincaré-Birkhoff Theorem on area-preserving twist maps to area-preserving twist maps $F$ that are random with respect to an ergodic probability measure. In this direction, we will prove several theorems concerning existence, density, and type of the fixed points. To this end first we introduce a randomized version of generalized generating functions, and verify the correspondence between its critical points and the fixed points of $F$, a fact which we successively exploit in order to prove the theorems. The study we carry out needs to combine probabilistic techniques with methods from nonlinear PDE, and from differential geometry, notably Moser’s method and Conley-Zehnder theory. Our stochastic model in the periodic case coincides with the classical setting of the Poincaré-Birkhoff Theorem.

References

Robert J. Adler and Jonathan E. Taylor, Random fields and geometry, Springer Monographs in Mathematics, Springer, New York, 2007. MR 2319516
Peter Albers, Joel W. Fish, Urs Frauenfelder, Helmut Hofer, and Otto van Koert, Global surfaces of section in the planar restricted 3-body problem, Arch. Ration. Mech. Anal. 204 (2012), no. 1, 273–284. MR 2898741, DOI 10.1007/s00205-011-0475-2
V. I. Arnol′d, Mathematical methods of classical mechanics, Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 0690288, DOI 10.1007/978-1-4757-1693-1
V. I. Arnol′d and A. Avez, Ergodic problems of classical mechanics, W. A. Benjamin, Inc., New York-Amsterdam, 1968. Translated from the French by A. Avez. MR 0232910
Michèle Audin, Vladimir Igorevich Arnold and the invention of symplectic topology, Contact and symplectic topology, Bolyai Soc. Math. Stud., vol. 26, János Bolyai Math. Soc., Budapest, 2014, pp. 1–25. MR 3220939, DOI 10.1007/978-3-319-02036-5_{1}
Jean-Marc Azaïs and Mario Wschebor, Level sets and extrema of random processes and fields, John Wiley & Sons, Inc., Hoboken, NJ, 2009. MR 2478201, DOI 10.1002/9780470434642
June Barrow-Green, Poincaré and the three body problem, History of Mathematics, vol. 11, American Mathematical Society, Providence, RI; London Mathematical Society, London, 1997. MR 1415387, DOI 10.1090/hmath/011
George D. Birkhoff, Proof of Poincaré’s geometric theorem, Trans. Amer. Math. Soc. 14 (1913), no. 1, 14–22. MR 1500933, DOI 10.1090/S0002-9947-1913-1500933-9
George D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), no. 2, 199–300. MR 1501070, DOI 10.1090/S0002-9947-1917-1501070-3
George D. Birkhoff, An extension of Poincaré’s last geometric theorem, Acta Math. 47 (1926), no. 4, 297–311. MR 1555218, DOI 10.1007/BF02559515
George D. Birkhoff, Dynamical systems, American Mathematical Society Colloquium Publications, Vol. IX, American Mathematical Society, Providence, R.I., 1966. With an addendum by Jurgen Moser. MR 0209095
B. Bramham and H. Hofer, First steps towards a symplectic dynamics, Surveys in differential geometry. Vol. XVII, Surv. Differ. Geom., vol. 17, Int. Press, Boston, MA, 2012, pp. 127–177. MR 3076060, DOI 10.4310/SDG.2012.v17.n1.a3
M. Brown and W. D. Neumann, Proof of the Poincaré-Birkhoff fixed point theorem, Michigan Math. J. 24 (1977), no. 1, 21–31. MR 448339, DOI 10.1307/mmj/1029001816
Alexander D. Bruno, The restricted 3-body problem: plane periodic orbits, De Gruyter Expositions in Mathematics, vol. 17, Walter de Gruyter & Co., Berlin, 1994. Translated from the Russian by Bálint Érdi; With a preface by Victor G. Szebehely. MR 1301328, DOI 10.1515/9783110901733
Patricia H. Carter, An improvement of the Poincaré-Birkhoff fixed point theorem, Trans. Amer. Math. Soc. 269 (1982), no. 1, 285–299. MR 637039, DOI 10.1090/S0002-9947-1982-0637039-0
Marc Chaperon, Une idée du type “géodésiques brisées” pour les systèmes hamiltoniens, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 13, 293–296 (French, with English summary). MR 765426
Marc Chaperon, An elementary proof of the Conley-Zehnder theorem in symplectic geometry, Dynamical systems and bifurcations (Groningen, 1984) Lecture Notes in Math., vol. 1125, Springer, Berlin, 1985, pp. 1–8. MR 798078, DOI 10.1007/BFb0075631
Marc Chaperon, Recent results in symplectic geometry, Dynamical systems and ergodic theory (Warsaw, 1986) Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 143–159. MR 1102710
C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol′d, Invent. Math. 73 (1983), no. 1, 33–49. MR 707347, DOI 10.1007/BF01393824
Wei Yue Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc. 88 (1983), no. 2, 341–346. MR 695272, DOI 10.1090/S0002-9939-1983-0695272-2
J. L. Doob, Random processes, John Wiley and Sons, Inc., New York; Chapman and Hall, Limited, London, 1953. viii+654 pp.
D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes, Springer Series in Statistics, Springer-Verlag, New York, 1988. MR 950166
Albert Einstein, Investigations on the theory of the Brownian movement, Dover Publications, Inc., New York, 1956. Edited with notes by R. Fürth; Translated by A. D. Cowper. MR 0077443
Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228
Andreas Floer, Witten’s complex and infinite-dimensional Morse theory, J. Differential Geom. 30 (1989), no. 1, 207–221. MR 1001276
Andreas Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), no. 4, 575–611. MR 987770, DOI 10.1007/BF01260388
Andreas Floer, Elliptic methods in variational problems, ICM-90, Mathematical Society of Japan, Tokyo; distributed outside Asia by the American Mathematical Society, Providence, RI, 1990. A plenary address presented at the International Congress of Mathematicians held in Kyoto, August 1990. MR 1126912
John Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math. (2) 128 (1988), no. 1, 139–151. MR 951509, DOI 10.2307/1971464
John Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8$^*$ (1988), no. Charles Conley Memorial Issue, 99–107. MR 967632, DOI 10.1017/S0143385700009366
John Franks, Erratum to: “Generalizations of the Poincaré-Birkhoff theorem” [Ann. of Math. (2) 128 (1988), no. 1, 139–151; MR0951509], Ann. of Math. (2) 164 (2006), no. 3, 1097–1098. MR 2259255, DOI 10.4007/annals.2006.164.1097
Joseph Galante and Vadim Kaloshin, Destruction of invariant curves in the restricted circular planar three-body problem by using comparison of action, Duke Math. J. 159 (2011), no. 2, 275–327. MR 2824484, DOI 10.1215/00127094-1415878
Christophe Golé, Symplectic twist maps, Advanced Series in Nonlinear Dynamics, vol. 18, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. Global variational techniques. MR 1992005, DOI 10.1142/9789812810762
Lucien Guillou, Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff, Topology 33 (1994), no. 2, 331–351 (French). MR 1273787, DOI 10.1016/0040-9383(94)90016-7
Helmut Hofer, Lagrangian embeddings and critical point theory, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 6, 407–462. MR 831040, DOI 10.1016/S0294-1449(16)30394-8
H. Hofer, Arnold and symplectic geometry, Notices Amer. Math. Soc. 59 (2012), 499–502.
H. Hofer and D. A. Salamon, Floer homology and Novikov rings, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 483–524. MR 1362838
H. Hofer, K. Wysocki, and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2) 157 (2003), no. 1, 125–255. MR 1954266, DOI 10.4007/annals.2003.157.125
Helmut Hofer and Eduard Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. MR 1306732, DOI 10.1007/978-3-0348-8540-9
Howard Jacobowitz, Periodic solutions of $x^{\prime \prime }+f(x,t)=0$ via the Poincaré-Birkhoff theorem, J. Differential Equations 20 (1976), no. 1, 37–52. MR 393673, DOI 10.1016/0022-0396(76)90094-2
Howard Jacobowitz, Corrigendum: The existence of the second fixed point: a correction to “Periodic solutions of $x''+f(x, t)=0$ via the Poincaré-Birkhoff theorem” (J. Differential Equations 20 (1976), no. 1, 37–52), J. Differential Equations 25 (1977), no. 1, 148–149. MR 437857, DOI 10.1016/0022-0396(77)90187-5
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
A. N. Kolmogorov, Théorie générale des systèmes dynamiques et mécanique classique, Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, Vol. 1, Erven P. Noordhoff N. V., Groningen; North-Holland Publishing Co., Amsterdam, 1957, pp. 315–333 (French). MR 0097598
Patrice Le Calvez, Propriétés dynamiques des difféomorphismes de l’anneau et du tore, Astérisque 204 (1991), 131 (French, with English and French summaries). MR 1183304
Patrice Le Calvez and Jian Wang, Some remarks on the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc. 138 (2010), no. 2, 703–715. MR 2557187, DOI 10.1090/S0002-9939-09-10105-3
Gang Liu and Gang Tian, Floer homology and Arnold conjecture, J. Differential Geom. 49 (1998), no. 1, 1–74. MR 1642105
John N. Mather, Dynamics of area preserving maps, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 1190–1194. MR 934323
Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. MR 1698616
Jürgen Moser, Stable and random motions in dynamical systems, Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. MR 0442980
Edward Nelson, Dynamical theories of Brownian motion, Princeton University Press, Princeton, N.J., 1967. MR 0214150, DOI 10.1515/9780691219615
Walter D. Neumann, Generalizations of the Poincaré Birkhoff fixed point theorem, Bull. Austral. Math. Soc. 17 (1977), no. 3, 375–389. MR 584597, DOI 10.1017/S0004972700010650
Kaoru Ono, On the Arnol′d conjecture for weakly monotone symplectic manifolds, Invent. Math. 119 (1995), no. 3, 519–537. MR 1317649, DOI 10.1007/BF01245191
H. Poincaré, Les méthodes nouvelles de la mécanique céleste. Tome III, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Librairie Scientifique et Technique Albert Blanchard, Paris, 1987 (French). Invariant intégraux. Solutions périodiques du deuxième genre. Solutions doublement asymptotiques. [Integral invariants. Periodic solutions of the second kind. Doubly asymptotic solutions]; Reprint of the 1899 original; Bibliothèque Scientifique Albert Blanchard. [Albert Blanchard Scientific Library]. MR 926908
H. Poincaré, Sur un théorème de géométrie, Rend. Circ. Mat. Palermo 33 (1912), 375–407.
Leonid Polterovich, The geometry of the group of symplectic diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. MR 1826128, DOI 10.1007/978-3-0348-8299-6
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
Claude Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), no. 4, 685–710. MR 1157321, DOI 10.1007/BF01444643
Alan Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), no. 3, 523–557. MR 723816
Alan Weinstein, On extending the Conley-Zehnder fixed point theorem to other manifolds, Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 541–544. MR 843640
E. Zehnder, The Arnol′d conjecture for fixed points of symplectic mappings and periodic solutions of Hamiltonian systems, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 1237–1246. MR 934328