Abstract
We show that a generic area-preserving two-dimensional map with an elliptic periodic point is C ω-universal, i.e., its renormalized iterates are dense in the set of all real-analytic symplectic maps of a two-dimensional disk. The results naturally extend onto Hamiltonian and volume-preserving flows.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Zehnder, E., Homoclinic Points Near Elliptic Fixed Points, Comm. Pure Appl. Math., 1973, vol. 26, pp. 131–182.
Turaev, D., Polynomial Approximations of Symplectic Dynamics and Richness of Chaos in Non-Hyperbolic Area-Preserving Maps, Nonlinearity, 2003, vol. 16, pp. 123–135.
Gonchenko, S., Turaev, D., and Shilnikov, L., Homoclinic Tangencies of Arbitrarily High Orders in Conservative and Dissipative Two-Dimensional Maps, Nonlinearity, 2007, vol. 20, pp. 241–275.
Mora, L. and Romero, N., Persistence of Homoclinic Tangencies for Area-Preserving Maps, Ann. Fac. Sci. Toulouse (6), 1997, vol. 6, no. 4, pp. 711–725.
Broer, H. and Tangerman, F., From a Differentiable to a Real Analytic Perturbation Theory, Applications to the Kupka-Smale Theorem, Ergodic Theory Dynam. Systems, 1986, vol. 6, pp. 345–362.
Gelfreich, V. and Lazutkin, V., Splitting of Separatrices: Perturbation Theory and Exponential Smallness, Russian Math. Surveys, 2001, vol. 56, pp. 499–558.
Duarte, P., Elliptic Isles in Families of Area-Preserving Maps, Ergodic Theory Dynam. Systems, 2008, vol. 28, no. 6, pp. 1781–1813.
Newhouse, S., The Abundance of Wild Hyperbolic Sets and Nonsmooth Stable Sets for Diffeomorphisms, Inst. Hautes’ Etudes Sci. Publ. Math., 1979, vol. 50, pp. 101–151.
Newhouse, S., Quasi-Elliptic Periodic Points in Conservative Dynamical Systems, Amer. J. Math., 1977, vol. 99, pp. 1061–1087.
Lerman, L., Complex Dynamics and Bifurcations in a Hamiltonian System Having a Transversal Homoclinic Orbit to a Saddle Focus, Chaos, 1991, vol. 1, no. 2, pp. 174–180.
Biragov, V. and Shilnikov, L., On the Bifurcation of a Saddle-Focus Separatrix Loop in a Three-Dimensional Conservative Dynamical System, Selecta Math. Soviet., 1992, vol. 11, no. 4, pp. 333–340.
Gonchenko, S. and Shilnikov, L., On Two-Dimensional Area-Preserving Diffeomorphisms with Infinitely Many Elliptic Islands, J. Stat. Phys., 2000, vol. 101, pp. 321–356.
Duarte, P., Abundance of Elliptic Isles at Conservative Bifurcations, Dynam. Stability Systems, 1999, vol. 14, no. 4, pp. 339–356.
Neishtadt, A., Simó, C., Treschev, D., and Vasiliev A., Periodic Orbits and Stability Islands in Chaotic Seas Created by Separatrix Crossings in Slow-Fast Systems, Discrete Contin. Dyn. Syst. Ser. B, 2008, vol. 10, pp. 621–650.
Turaev, D. and Rom-Kedar, V., Islands Appearing in Near-Ergodic Flows, Nonlinearity, 1998, vol. 11, pp. 575–600.
Rom-Kedar, V. and Turaev, D., Big Islands in Dispersing Billiard-Like Potentials, Phys. D, 1999, vol. 130, pp. 187–210.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gelfreich, V., Turaev, D. Universal dynamics in a neighborhood of a generic elliptic periodic point. Regul. Chaot. Dyn. 15, 159–164 (2010). https://doi.org/10.1134/S156035471002005X
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S156035471002005X