# Conjugation to a shift and the splitting of invariant manifolds

Applicationes Mathematicae (1997)

- Volume: 24, Issue: 2, page 127-140
- ISSN: 1233-7234

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topGelfreich, Vassiliĭ. "Conjugation to a shift and the splitting of invariant manifolds." Applicationes Mathematicae 24.2 (1997): 127-140. <http://eudml.org/doc/219157>.

@article{Gelfreich1997,

abstract = {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.},

author = {Gelfreich, Vassiliĭ},

journal = {Applicationes Mathematicae},

keywords = {normal form; separatrix splitting; finite-difference equation; diffeomorphism; invariant curve; uniform estimates; limit flow separatrix; parabolic fixed points},

language = {eng},

number = {2},

pages = {127-140},

title = {Conjugation to a shift and the splitting of invariant manifolds},

url = {http://eudml.org/doc/219157},

volume = {24},

year = {1997},

}

TY - JOUR

AU - Gelfreich, Vassiliĭ

TI - Conjugation to a shift and the splitting of invariant manifolds

JO - Applicationes Mathematicae

PY - 1997

VL - 24

IS - 2

SP - 127

EP - 140

AB - 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.

LA - eng

KW - normal form; separatrix splitting; finite-difference equation; diffeomorphism; invariant curve; uniform estimates; limit flow separatrix; parabolic fixed points

UR - http://eudml.org/doc/219157

ER -

## References

top- [Fon95] E. Fontich, Rapidly forced planar vector fields and splitting of separatrices, J. Differential Equations 119 (1995), 310-335. Zbl0827.34041
- [FS90] E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms, Ergodic. Theory Dynam. Systems 10 (1990), 295-318. Zbl0706.58061
- [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).

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