No CrossRef data available.
Article contents
Robust transitivity and domination for endomorphisms displaying critical points
Part of:
Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Hamiltonian and Lagrangian mechanics
Published online by Cambridge University Press: 14 April 2023
Abstract
We show that robustly transitive endomorphisms of a closed manifold must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result, we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation.
MSC classification
- Type
- Original Article
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press
References
Aoki, N. and Hiraide, K.. Topological Theory of Dynamical Systems: Recent Advances (North-Holland Mathematical Library, 52). Elsevier Science, Amsterdam, 1994.Google Scholar
Avila, A. and Bochi, J.. Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms. Trans. Amer. Math. Soc. 364(6) (2012), 2883–2907.10.1090/S0002-9947-2012-05423-7CrossRefGoogle Scholar
Berger, P. and Rovella, A.. On the inverse limit stability of endomorphisms. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(3) (2013), 463–475.10.1016/j.anihpc.2012.10.001CrossRefGoogle Scholar
Bochi, J. and Gourmelon, N.. Some characterizations of domination. Math. Z. 262(1) (2009), 221–231.10.1007/s00209-009-0494-yCrossRefGoogle Scholar
Bochi, J. and Morris, I. D.. Continuity properties of the lower spectral radius. Proc. Lond. Math. Soc. (3) 110(2) (2015), 477–509.10.1112/plms/pdu058CrossRefGoogle Scholar
Bochi, J., Potrie, R. and Sambarino, A.. Anosov representations and dominated splittings. J. Eur. Math. Soc. (JEMS) 21(11) (2019), 3343–3414.10.4171/jems/905CrossRefGoogle Scholar
Bonatti, C. and Díaz, L. J.. Persistent nonhyperbolic transitive diffeomorphisms. Ann. of Math. (2) 143(2) (1996), 357–396.10.2307/2118647CrossRefGoogle Scholar
Bonatti, C., Díaz, L. J. and Pujals, E. R.. A
${C}^1$
-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. (2) 158(2) (2003), 355–418.10.4007/annals.2003.158.355CrossRefGoogle Scholar
Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115(1) (2000), 157–193.10.1007/BF02810585CrossRefGoogle Scholar
Crovisier, S. and Potrie, R.. Introduction to partially hyperbolic dynamics. School on Dynamical Systems. Vol. 3, no. 1. ICTP, Trieste, 2015, pp. 1–71.Google Scholar
Díaz, L. J., Pujals, E. R. and Ures, R.. Partial hyperbolicity and robust transitivity. Acta Math. 183(1) (1999), 1–43.10.1007/BF02392945CrossRefGoogle Scholar
Franks, J.. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. 158(2) (1971), 301–308.10.1090/S0002-9947-1971-0283812-3CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R.. Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge, 2012.10.1017/CBO9781139020411CrossRefGoogle Scholar
Iglesias, J., Lizana, C. and Portela, A.. Robust transitivity for endomorphisms admitting critical points. Proc. Amer. Math. Soc. 144(3) (2016), 1235–1250.10.1090/proc/12799CrossRefGoogle Scholar
Iglesias, J. and Portela, A.. An example of a map which is
${C}^2$
-robustly transitive but not
${C}^1$
-robustly transitive. Colloq. Math. 152(2) (2018), 285–297.10.4064/cm7131-5-2017CrossRefGoogle Scholar
Lizana, C. and Pujals, E.. Robust transitivity for endomorphisms. Ergod. Th. & Dynam. Sys. 33(4) (2013), 1082–1114.10.1017/S0143385712000247CrossRefGoogle Scholar
Lizana, C. and Ranter, W.. Topological obstructions for robustly transitive endomorphisms on surfaces. Adv. Math. 390 (2017), 107901.10.1016/j.aim.2021.107901CrossRefGoogle Scholar
Lizana, C. and Ranter, W.. New classes of
${C}^1$
-robustly transitive maps with persistent critical points. Preprint, 2020, arXiv:1902.06781.Google Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17(4) (1978), 383–396.10.1016/0040-9383(78)90005-8CrossRefGoogle Scholar
Mañé, R.. An ergodic closing lemma. Ann. of Math. (2) 116(3) (1982), 503–540.10.2307/2007021CrossRefGoogle Scholar
Milnor, J. W. and Stasheff, J. D.. Characteristic Classes (Annals of Mathematics Studies, 76). Princeton University Press, Princeton, NJ, 1974.10.1515/9781400881826CrossRefGoogle Scholar
Morelli, J. C.. An example of a
${T}^n$
endomorphism that is persistently singular and
$C^1$
robustly transitive. Bull. Soc. Math. France 149(3) (2021), 501–519.Google Scholar
Potrie, R.. Partial hyperbolicity and attracting regions in 3-dimensional manifolds. Preprint, 2012, arXiv:1207.1822.Google Scholar
Quas, A., Thieullen, P. and Zarrabi, M.. Explicit bounds for separation between Oseledets subspaces. Dyn. Syst. 34(3) (2019), 517–560.10.1080/14689367.2019.1571562CrossRefGoogle Scholar
Shub, M.. Topologically transitive diffeomorphisms of
${T}^4$
. Proc. Symp. on Differential Equations and Dynamical Systems (Lecture Notes in Mathematics, 206). Ed. Chillingworth, D.. Springer, Berlin, 1971.Google Scholar
Zhu, F.. Relatively dominated representations. Ann. Inst. Fourier 71(5) (2021), 2169–2235.10.5802/aif.3449CrossRefGoogle Scholar