Hostname: page-component-cb9f654ff-5kfdg Total loading time: 0 Render date: 2025-08-29T09:29:41.625Z Has data issue: false hasContentIssue false
  • English
  • Français

High-order homoclinic tangencies of corank 2

Part of: Smooth dynamical systems: general theory Low-dimensional dynamical systems Local and nonlocal bifurcation theory

Published online by Cambridge University Press:  29 August 2025

DMITRII MINTS
DMITRII MINTS*
Affiliation:
https://ror.org/041kmwe10 Imperial College London , SW7 2AZ London, UK and National Research University Higher School of Economics, 25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia
*
e-mail: d.mints22@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that in the space of $C^r$ maps $(r=2,\ldots ,\infty ,\omega )$ of a smooth manifold of dimension at least 4, there exist open regions where maps with infinitely many corank-2 homoclinic tangencies of all orders are dense. The result is applied to show the existence of maps with universal two-dimensional dynamics, that is, maps whose iterations approximate the dynamics of every map of a two-dimensional disk with an arbitrarily good accuracy. We show that maps with universal two-dimensional dynamics are $C^r$-generic in the regions under consideration.

Keywords

homoclinic tangencyNewhouse domainuniversal dynamics

MSC classification

Primary: 37C29: Homoclinic and heteroclinic orbits 37G25: Bifurcations connected with nontransversal intersection 37C20: Generic properties, structural stability 37E20: Universality, renormalization

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 45
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abraham, R. and Smale, S.. Nongenericity of $\varOmega$ -stability. Proc. Sympos. Pure Math. 14 (1970), 5–8.10.1090/pspum/014/0271986CrossRefGoogle Scholar
Arnold, V. I., Afraimovich, V. S., Ilyashenko, Yu. S. and Shilnikov, L. P.. Bifurcation theory. Dynamical Systems – 5 (Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 5). VINITI, Moscow, 1986 (in Russian). Translated in Encyclopedia of Math. Sci., Vol. 5. Springer-Verlag, 1994.Google Scholar
Arnold, V. I., Gusein-Zade, S. M. and Varchenko, A. N.. Singularities of Differentiable Maps, Volume 1. Springer Science & Business Media, New York, 2012.Google Scholar
Asaoka, M.. Hyperbolic sets exhibiting ${C}^1$ -persistent homoclinic tangency for higher dimensions. Proc. Amer. Math. Soc. 136(2) (2008), 677–686.10.1090/S0002-9939-07-09115-0CrossRefGoogle Scholar
Asaoka, M.. Abundance of fast growth of the number of periodic points in 2-dimensional area-preserving dynamics. Comm. Math. Phys. 356 (2017), 1–17.10.1007/s00220-017-2972-0CrossRefGoogle Scholar
Asaoka, M.. Stable intersection of Cantor sets in higher dimension and robust homoclinic tangency of the largest codimension. Trans. Amer. Math. Soc. 375(2) (2022), 873–908.CrossRefGoogle Scholar
Asaoka, M., Shinohara, K. and Turaev, D.. Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics. Math. Ann. 368 (2017), 1277–1309.10.1007/s00208-016-1468-0CrossRefGoogle Scholar
Barrientos, P. G. and Raibekas, A.. Robust tangencies of large codimension. Nonlinearity 30 (2017), 4369–4409.CrossRefGoogle Scholar
Barrientos, P. G. and Raibekas, A.. Robustly non-hyperbolic transitive symplectic dynamics. Discrete Contin. Dyn. Syst. 38(12) (2018), 5993–6013.10.3934/dcds.2018259CrossRefGoogle Scholar
Barrientos, P. G. and Raibekas, A.. Robust degenerate unfoldings of cycles and tangencies. J. Dynam. Differential Equations 33(1) (2021), 177–209.10.1007/s10884-020-09857-0CrossRefGoogle Scholar
Berger, P.. Generic family displaying robustly a fast growth of the number of periodic points. Acta Math. 227(2) (2021), 205–262.CrossRefGoogle Scholar
Berger, P., Florio, A. and Peralta-Salas, D.. Steady Euler flows on ${\mathbb{R}}^3$ with wild and universal dynamics. Comm. Math. Phys. 401 (2023), 1–47.10.1007/s00220-023-04660-6CrossRefGoogle Scholar
Bonatti, C. and Díaz, L.. Abundance of ${C}^1$ -robust homoclinic tangencies. Trans. Amer. Math. Soc. 364(10) (2012), 5111–5148.10.1090/S0002-9947-2012-05445-6CrossRefGoogle Scholar
Bonatti, C. and Díaz, L. J.. Robust heterodimensional cycles and generic dynamics. J. Inst. Math. Jussieu 7(3) (2008), 469–525.10.1017/S1474748008000030CrossRefGoogle Scholar
Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102). Springer Berlin, Heidelberg, 2004.Google Scholar
Bowen, R.. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377–397.Google Scholar
Broer, H. W. and Tangerman, F. M.. From a differentiable to a real analytic perturbation theory, applications to the Kupka Smale theorems. Ergod. Th. & Dynam. Sys. 6(3) (1986), 345–362.10.1017/S0143385700003540CrossRefGoogle Scholar
Buzzi, J., Crovisier, S. and Fisher, T.. The entropy of ${C}^1$ -diffeomorphisms without a dominated splitting. Trans. Amer. Math. Soc. 370(9) (2018), 6685–6734.10.1090/tran/7380CrossRefGoogle Scholar
Catalan, T.. A link between topological entropy and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 39(3) (2019), 620–637.10.1017/etds.2017.39CrossRefGoogle Scholar
Crovisier, S. and Pujals, E. R.. Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms. Invent. Math. 201(2) (2015), 385–517.10.1007/s00222-014-0553-9CrossRefGoogle Scholar
Domitrz, W., Mormul, P. and Pragacz, P.. Order of tangency between manifolds. Schubert Calculus and Its Applications in Combinatorics and Representation Theory. Ed. J. Hu, C. Li and L. C. Mihalcea. Springer, Singapore, 2017, pp. 27–42.Google Scholar
Gonchenko, S., Turaev, D. and Shilnikov, L.. Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps. Nonlinearity 20(2) (2007), 241–275.CrossRefGoogle Scholar
Gonchenko, S. V., Gonchenko, V. S. and Tatjer, J. C.. Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps. Regul. Chaotic Dyn. 12 (2007), 233–266.Google Scholar
Gonchenko, S. V., Ovsyannikov, I. I. and Tatjer, J. C.. Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points. Regul. Chaotic Dyn. 19 (2014), 495–505.10.1134/S1560354714040054CrossRefGoogle Scholar
Gonchenko, S. V. and Shilnikov, L. P. (eds.). Homoclinic Tangencies. RCD, Moscow and Izhevsk, Russia, 2007 (in Russian).Google Scholar
Gonchenko, S. V., Shilnikov, L. P. and Turaev, D. V.. On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: I. Nonlinearity 21(5) (2008), 923–972.10.1088/0951-7715/21/5/003CrossRefGoogle Scholar
Gonchenko, S. V., Turaev, D. V. and Shilnikov, L. P.. On the existence of Newhouse domains in a neighborhood of systems with a structurally unstable Poincaré homoclinic curve (the higher-dimensional case). Russian Acad. Sci. Dokl. Math. 47(2) (1993), 268–273.Google Scholar
Gonchenko, S. V., Turaev, D. V. and Shilnikov, L. P.. Homoclinic tangencies of an arbitrary order in Newhouse domains. Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz. 67 (1999), 69–128 (in Russian); Engl. Transl. J. Math. Sci. (N.Y.) 105(1) (2001), 1738–1778.Google Scholar
Gourmelon, N., Li, D., Mints, D. and Turaev, D.. Stabilization of homoclinic tangencies, to appear.Google Scholar
Kaloshin, V. Y.. Generic diffeomorphisms with superexponential growth of number of periodic orbits. Comm. Math. Phys. 211 (2000), 253–271.Google Scholar
Kuznetsov, Yu. A.. Elements of Applied Bifurcation Theory. Springer Cham, Switzerland, 2023.CrossRefGoogle Scholar
Li, D.. ${C}^1$ -robust homoclinic tangencies. Preprint, 2024, arXiv:2406.12500.Google Scholar
Li, D. and Turaev, D.. Persistence of heterodimensional cycles. Invent. Math. 236(3) (2024), 1413–1504.10.1007/s00222-024-01255-3CrossRefGoogle Scholar
Mints, D. and Turaev, D.. Highly degenerate homoclinic tangencies and universal dynamics in the Newhouse domain, to appear.Google Scholar
Newhouse, S. E.. Non-density of axiom A on ${S}^2$ . Proc. Sympos. Pure Math. 14 (1970), 191–202.10.1090/pspum/014/0277005CrossRefGoogle Scholar
Newhouse, S. E.. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 101–151.10.1007/BF02684771CrossRefGoogle Scholar
Newhouse, S. E., Palis, J. and Takens, F.. Bifurcations and stability of families of diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 57 (1983), 5–71.10.1007/BF02698773CrossRefGoogle Scholar
Palis, J.. On the ${C}^1$ $\varOmega$ -stability conjecture. Publ. Math. Inst. Hautes Études Sci. 66 (1987), 211–215.10.1007/BF02698932CrossRefGoogle Scholar
Palis, J. and Takens, F.. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Cambridge Studies in Advanced Mathematics, 35). Cambridge University Press, Cambridge, UK, 1993.Google Scholar
Palis, J. and Viana, M.. High dimension diffeomorphisms displaying infinitely many periodic attractors. Ann. of Math. (2) 140(1) (1994), 207–250.10.2307/2118546CrossRefGoogle Scholar
Pujals, E. R. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. (2) 151(3) (2000), 961–1023.Google Scholar
Romero, N.. Persistence of homoclinic tangencies in higher dimensions. Ergod. Th. & Dynam. Sys. 15(4) (1995), 735–757.10.1017/S0143385700008634CrossRefGoogle Scholar
Shilnikov, L. P.. On a Poincaré–Birkhoff problem. Math. USSR-Sb. 3(3) (1967), 353–371.CrossRefGoogle Scholar
Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. and Chua, L. O.. Methods of Qualitative Theory in Nonlinear Dynamics, Part I. World Scientific, Singapore, 1998.10.1142/9789812798596CrossRefGoogle Scholar
Simon, C. P.. Instability in $\mathit{Diff}^r\left({T}^3\right)$ and the nongenericity of rational zeta functions. Trans. Amer. Math. Soc. 174 (1972), 217–242.Google Scholar
Smale, S.. Diffeomorphisms with many periodic points. Differential and Combinatorial Topology. Ed. S. S. Cairns. Princeton University Press, Princeton, NJ, 1965, pp. 63–80.10.1515/9781400874842-006CrossRefGoogle Scholar
Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73(6) (1967), 747–817.Google Scholar
Smale, S.. The $\varOmega$ -stability theorem. Proc. Sympos. Pure Math. 14 (1970), 289–297.Google Scholar
Tatjer, J. C.. Three-dimensional dissipative diffeomorphisms with homoclinic tangencies. Ergod. Th. & Dynam. Sys. 21(1) (2001), 249–302.10.1017/S0143385701001146CrossRefGoogle Scholar
Turaev, D.. On dimension of non-local bifurcational problems. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6(5) (1996), 919–948.Google Scholar
Turaev, D.. Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps. Nonlinearity 16(1) (2002), 123–135.10.1088/0951-7715/16/1/308CrossRefGoogle Scholar
Turaev, D.. Diffeomorphisms which cannot be topologically conjugate to diffeomorphisms of a higher smoothness. Ergod. Th. & Dynam. Sys. 31 (2011), 563–569.10.1017/S014338570900114XCrossRefGoogle Scholar
Turaev, D.. Maps close to identity and universal maps in the Newhouse domain. Comm. Math. Phys. 335 (2015), 1235–1277.Google Scholar
Turaev, D.. Richness of chaos in the absolute Newhouse domain. Proceedings of the International Congress of Mathematicians 2010 (ICM 2010). Ed. R. Bhatia. World Scientific, Singapore, 2010, pp. 1804–1815.Google Scholar