+7 (3412) 91 60 92

## Archive of Issues

Russia Saint Petersburg
Year
2013
Issue
4
Pages
27-35
 Section Mathematics Title Stable periodic points for smooth diffeomorphisms of multidimensional space Author(-s) Vasil'eva E.V.a Affiliations Saint Petersburg State Universitya Abstract We regard $C^r$-smooth ($r \geqslant 1$) self-diffeomorphism of multidimensional space with a hyperbolic fixed point and non-transversal homoclinic point. In the works by Sh. Newhouse, L.P. Shil'nikov, B.F. Ivanov and other authors it is shown that under certain condition on the type of contact of stable and unstable manifolds, the neighborhoods of the homoclinic point may contain a countable set of stable periodic points, but at least one of their characterictic exponents tends to zero with the increase of a period. The goal of this work is to prove that under certain conditions imposed on the character of tangency between the stable and unstable manifolds, the neighborhood of the homoclinic point may contain an infinite set of stable periodic points whose characteristic exponents are negative and bounded away from zero. Keywords diffeomorphism of multidimentional space, homoclinic points, stable periodic points UDC 517.925.53 MSC 37C29, 37C75 DOI 10.20537/vm130404 Received 18 November 2013 Language Russian Citation Vasil'eva E.V. Stable periodic points for smooth diffeomorphisms of multidimensional space, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 4, pp. 27-35. References Ivanov B.F. Stability of the trajectories that do not leave the neighborhood of a homoclinic curve, Differ. Uravn., 1979, vol. 15, no. 8, pp. 1411-1414. Gonchenko S.V., Turaev D.V., Shil'nikov L.P. Dynamical phenomena in multidimensional systems with a structurally unstable homoclinic Poincare' curve, Russian Academy of Sciences. Doklady. Mathematics, 1993, vol. 47, no. 3, pp. 410-415. Newhouse Sh. Diffeomorphisms with infinitely many sinks, Topology, 1973, vol. 12, pp. 9-18. Vasil'eva E.V. Stable nonperiodic points of two-dimensional $C$$1$ -diffeomorphisms, Vestnik St. Petersburg University: Mathematics, 2007, vol. 40, issue 2, pp. 107-113. Vasil'eva E.V. Smooth diffeomorphisms of the plane with stable periodic points in a neighborhood of a homoclinic point, Differential Equations, 2012, vol. 48, no. 10, pp. 1335-1340. Pliss V.A. Integral'nye mnozhestva periodicheskikh system differentsial'nykh uravnenii (Integral sets of periodical systems of differential equations), Moscow: Nauka, 1977, 304 p. Vasil'eva E.V. Diffeomorphisms of multidimensional space with infinite set of stable periodic points, Vestnik St. Petersburg University: Mathematics, 2012, vol. 45, issue 3, pp. 115-124. Full text