Conjugation to a shift and the splitting of invariant manifolds

• Autorzy: Vassiliĭ G. Gelfreich
• Języki publikacji: EN

Abstrakty

EN

We give sufficient conditions for a diffeomorphism in the plane to be analytically conjugate to a shift in a complex neighborhood of a segment of an invariant curve. For a family of functions close to the identity uniform estimates are established. As a consequence an exponential upper estimate for splitting of separatrices is established for diffeomorphisms of the plane close to the identity. The constant in the exponent is related to the width of the analyticity domain of the limit flow separatrix. Unlike the previous works the cases of non-area-preserving maps and parabolic fixed points are included.

Słowa kluczowe

EN

normal form; separatrix splitting; finite-difference equation

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 1996-1997
• Tom: 24
• Numer: 2
• Strony: 127-140

Daty

wydano

1997

otrzymano

1995-05-29

poprawiono

1995-12-07

Twórcy

autor

Vassiliĭ G. Gelfreich

• Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via, 585, 08071 Barcelona, Spain

Bibliografia

• [Fon95] E. Fontich, Rapidly forced planar vector fields and splitting of separatrices, J. Differential Equations 119 (1995), 310-335.
• [FS90] E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms, Ergodic. Theory Dynam. Systems 10 (1990), 295-318.
• [Laz84] V. F. Lazutkin, Splitting of separatrices for Chirikov's standard map, VINITI no. 6372/84, 1984 (in Russian).
• [Laz87] V. F. Lazutkin, Separatrices splitting for a standard family of the area-preserving maps, in: M. Sh. Birman (ed.), Wave Propagation. Scattering Theory, Topics in Math. Phys. 12, Leningrad State University, 1987, 32-41 (in Russian).
• [Laz91] V. F. Lazutkin, Exponential splitting of separatrices and an analytical integral for the semistandard map, preprint, Université Paris VII, 1991.
• [Nei84] A. I. Neishtadt, The separation of motion in systems with rapidly rotating phase, Prikl. Mat. Mekh. 48 (1984), 197-204, (in Russian).