Contraction method and lambda-lemma. (English) Zbl 1369.37020

Weber, Joa

São Paulo J. Math. Sci. 9, No. 2, 263-298 (2015).

Summary: We reprove the \(\lambda\)-Lemma for finite dimensional gradient flows by generalizing the well-known contraction method proof of the local (un)stable manifold theorem. This only relies on the forward Cauchy problem. We obtain a rather quantitative description of (un)stable foliations which allows to equip each leaf with a copy of the flow on the central leaf – the local (un)stable manifold. These dynamical thickenings are key tools in the author’s recent work [Topol. Methods Nonlinear Anal. 47, No. 2, 641–646 (2016; Zbl 1373.37042)]. The present paper provides their construction.

MSC:

37B35	Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems
37B30	Index theory for dynamical systems, Morse-Conley indices
37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C85	Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)

Keywords:

hyperbolic dynamics; lambda-lemma; invariant foliations

Citations:

Zbl 1373.37042

References:

[1]	Agarwal, R.P., Lakshmikantham, V.: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, Volume 6 of Series in Real Analysis. World Scientific Publishing Co., Inc., River Edge, NJ (1993) · Zbl 0785.34003 · doi:10.1142/1988
[2]	Burghelea, D., Friedlander, L., Kappeler, T.: On the Space of Trajectories of a Generic Vector Field (2011). arXiv:1101.0778 · Zbl 1240.37016
[3]	Banyaga, A., Hurtubise, D.: Lectures on Morse Homology, Volume 29 of Kluwer Texts in the Mathematical Sciences. Kluwer Academic Publishers Group, Dordrecht (2004) · Zbl 1080.57001
[4]	Bott, R.: Morse theory indomitable. Inst. Hautes Études Sci. Publ. Math. 68, 99-114 (1988) · Zbl 0685.58009 · doi:10.1007/BF02698544
[5]	Chow, S.N., Hale, J.K.: Methods of Bifurcation Theory, Volume 251 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science]. Springer, Berlin (1982) · Zbl 0487.47039
[6]	Hadamard, J.: Sur l’itération et les solutions asymptotiques des équations différentielles. Bull. Soc. Math. Fr. 29, 224-228 (1901) · JFM 32.0314.01
[7]	Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Volume 840 of Lecture Notes in Mathematics. Springer, Berlin (1981) · Zbl 0456.35001
[8]	Hirsch, M.W.: Differential Topology. Springer, Heidelberg (1976). Graduate Texts in Mathematics, No. 33 · Zbl 0356.57001 · doi:10.1007/978-1-4684-9449-5
[9]	Jost, J.: Riemannian Geometry and Geometric Analysis. Universitext, 6th edn. Springer, Heidelberg (2011) · Zbl 1227.53001 · doi:10.1007/978-3-642-21298-7
[10]	Lawson Jr, H.B.: Foliations. Bull. Am. Math. Soc. 80, 369-418 (1974) · Zbl 0293.57014 · doi:10.1090/S0002-9904-1974-13432-4
[11]	Nicolaescu, L.: An Invitation to Morse Theory. Universitext, 2nd edn. Springer, New York (2011) · Zbl 1238.57001 · doi:10.1007/978-1-4614-1105-5
[12]	Palis Jr, J.: On Morse-Smale Diffeomorphisms. PhD thesis, UC Berkeley (1967) · Zbl 1306.37090
[13]	Palis Jr, J.: On Morse-Smale dynamical systems. Topology 8, 385-404 (1969) · Zbl 0189.23902 · doi:10.1016/0040-9383(69)90024-X
[14]	Palis Jr, J., de Melo, W.: Geometric Theory of Dynamical Systems. Springer, New York. An introduction. Translated from the Portuguese by A. K, Manning (1982) · Zbl 0491.58001
[15]	Perron, O.: Über Stabilität und asymptotisches Verhalten der integrale von differentialgleichungssystemen. Math. Z. 29, 129-160 (1928) · JFM 54.0456.04 · doi:10.1007/BF01180524
[16]	Pesin, Y.B.: Lectures on Partial Hyperbolicity and Stable Ergodicity. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2004) · Zbl 1098.37024
[17]	Qin, L.: An Application of Topological Equivalence to Morse Theory (2011). arXiv:1102.2838
[18]	Salamon, D.: Morse theory, the Conley index and Floer homology. Bull. Lond. Math. Soc. 22(2), 113-140 (1990) · Zbl 0709.58011 · doi:10.1112/blms/22.2.113
[19]	Shub, M.: Global Stability of Dynamical Systems. Springer, New York (1987). With the collaboration of Albert Fathi and Rémi Langevin, Translated from the French by Joseph Christy · Zbl 0606.58003
[20]	Salamon, D., Weber, J.: Floer homology and the heat flow. Geom. Funct. Anal. 16(5), 1050-1138 (2006) · Zbl 1118.53056 · doi:10.1007/s00039-006-0577-4
[21]	Teschl, G.: Ordinary Differential Equations and Dynamical Systems, Volume 140 of Graduate Studies in Mathematics. Online edition, authorized by American Mathematical Society, Providence, RI (2012) · Zbl 1263.34002
[22]	Weber, J.: Classical Morse theory revisited I—-backward \[\lambda\] λ-Lemma and homotopy type. Topol. Methods Nonlinear Anal. (to appear). arXiv:1410.0995 · Zbl 1373.37042
[23]	Weber, J.: The Morse-Witten complex via dynamical systems. Expo. Math. 24(2), 127-159 (2006) · Zbl 1134.37324 · doi:10.1016/j.exmath.2005.09.001
[24]	Weber, J.: Morse homology for the heat flow. Math. Z. 275(1-2), 1-54 (2013) · Zbl 1319.53104 · doi:10.1007/s00209-012-1121-x
[25]	Weber, J.: A backward \[\lambda\] λ-lemma for the forward heat flow. Math. Ann. 359(3-4), 929-967 (2014) · Zbl 1306.37090 · doi:10.1007/s00208-014-1026-6
[26]	Weber, J.: Stable Foliations and Semi-Flow Morse Homology. Submitted (2014). arXiv:1408.3842 · Zbl 1497.58014
[27]	Zehnder, E.: Lectures on Dynamical Systems. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich (2010) · Zbl 1195.37001