Abstract
We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace which satisfies certain nonresonance conditions, the map leaves invariant a smooth manifold tangent to this subspace. This manifold is as smooth as the map—when the smoothness is measured in appropriate scales—but is unique amongC L invariant manifolds, whereL depends only on the spectrum of the linearization or on some more general smoothness classes that we detail. We show that if the nonresonance conditions are not satisfied, a smooth invariant manifold need not exist, and we also establish smooth dependence on parameters. We also discuss some applications of these invariant manifolds and briefly survey related work.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Abraham and J. Robbin,Transversal Mappings and Flows (Benjamin, New York, 1967).
A. Banyaga, R. de la Llave, and C. E. Wayne, Cohomology equations and geometric versions of Sternberg linearization theorem [mp_arc preprint 94-135].
N. Bogoliubov and A. Mitropolskii,Asymptotic Methods in the Theory of Nonlinear Oscillations (Gordon and Breach, New York, 1960).
M. Le Bellac,Des phénomènes critiques aux champs de jauge (Intereditions/Ed. du CNRS, Paris, 1988).
P. Collet, J.-P. Eckmann, and H. Koch, Period-doubling bifurcations for families of maps on ℝn,J. Stat. Phys. 25:1–14 (1980).
P. Collet, J.-P. Eckmann, and O. E. Lanford III, Universal properties for maps of an interval,Commun. Math. Phys. 76:211–254 (1984).
J. Casasayas, E. Fontich, and A. Nunes, Invariant manifolds for a class of parabolic points,Nonlinearity 5:1193–1209 (1992).
X. Cabré and E. Fontich, Regularity and uniqueness of one dimensional invariant manifolds, University of Barcelona preprint.
A. C. D. van Enter, R. Fernandez, and A. D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory,J. Stat. Phys. 72:879–1178 (1993).
N. Fenichel, Persistence and smoothness of invariant manifolds for flows,Ind. Math. J. 21:193–226 (1971).
N. Fenichel, Asymptotic stability with rate conditions,Ind. Math. J. 23:1109–1137 (1973).
E. Fontich, Invariant manifolds corresponding to eigenvalues equal to one in Henon-like maps, University of Barcelona preprint.
K. Gawdezki, A. Kupianen, and B. Tirozzi, Renormalons: A dynamical systems approach,Nucl. Phys. B 257:610–628 (1985).
M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets,Proc. Symp. Pure Math. 14:133–163 (1970).
J. Hale,Ordinary Differential Equations (Wiley, New York, 1969).
M. C. Irwin, On the smoothness of the composition map,Q. J. Math. 23:113–133 (1972).
M. C. Irwin, On the stable manifold theorem,Bull. Lond. Math. Soc. 2:196–198 (1970).
M. C. Irwin, A new proof of the pseudostable manifold theorem,J. Lond. Math. Soc. 21:557–566 (1980).
M. Jiang, R. de la Llave, and Y. Pesin, On the integrability of intermediate distributions for Anosov diffeomorphisms,Ergodic Theory Dynam. Syst. 15:317–331 (1995).
N. Kurzweil, On approximation in real Banach spaces,Studia Math. 14:213–231 (1954).
T. Kato,Perturbation Theory for Linear Operators, (Springer-Verlag, Berlin, 1976).
A. Kelley, The stable, center-stable, center, center-unstable, unstable invariant manifold theorems [Appendix to ref. 1].
B. Li, N. Madras, and A. D. Sokal, Critical exponents, hyperscaling and universal amplitude ratios for two and three dimensional self-avoiding walks,J. Stat. Phys. 80:661–754 (1995).
R. de la Llave and C. E. Wayne, On Irwin's proof of the pseudostable manifold theorem,Math. Z. 219:301–321 (1995).
O. E. Lanford III, Bifurcation of periodic solutions into invariant tori: The work of Ruelle and Takens, inLecture Notes in Mathematics, Vol. 322 (Springer-Verlag, Berlin, 1973).
O. E. Lanford III, Introduction to the mathematical theory of dynamical systems, inChaotic Behavior of Deterministic Systems, Les Houches 1981 (North-Holland, 1983).
E. B. Leach and J. H. M. Whitfield, Differentiable functions and rough norms on Banach spaces,Proc. AMS 33:120–126 (1972).
J. Marsden and M. McCracken,The Hopf Bifurcation and Its Applications (Springer-Verlag, Berlin, 1976).
F. Martinelli and E. Olivieri, Some remarks on pathologies or renormalization group transformations for the Ising model,J. Stat. Phys. 72:1169–1178 (1993).
E. Nelson,Topics in Dynamics: I. Flows (Princeton University Press, Princeton, New Jersey, 1969).
Y. Pesin, Families of invariant manifolds corresponding to non-zero characteristic exponents,Math. USSR Izv. 10:1261–1305 (1976).
J. Pöschel, On invariant manifolds of complex analytic mappings near fixed points, inCritical Phenomena, Random Systems, Gauge Theories, K. Osterwalder and R. Stora, eds. (North-Holland, Amsterdam, 1986).
M. Reed and B. Simon,Methods of Modern Mathematical Physics I (Academic Press, New York, 1972).
S. Sternberg, On the structure of local homeomorphism of Euclideann-space II,Am. J. Math. 80:623–631 (1958).
E. B. Vul, Y. G. Sinai, and K. M. Khanin, Feigenbaum universality and the thermodynamic formalism,Russ. Math. Surv. 39:1–40 (1984).
S. Wiggins,Normally Hyperbolic Manifolds in Dynamical Systems (Springer-Verlag, Berlin, 1994).
M. Hirsch, C. Pugh, and M. Shub, Invariant manifolds (Springer-Verlag, New York),Lec. Notes in Math. 583 (1977).
M. Shub, Stabilité globale des systèmes dynamiques,Astérisque 56 (1978).
J. Mather, Characterization of Anosov diffeomorphisms,Indag. Mat. 30:479–483 (1968).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
de la Llave, R. Invariant manifolds associated to nonresonant spectral subspaces. J Stat Phys 87, 211–249 (1997). https://doi.org/10.1007/BF02181486
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02181486