The closing lemma and the planar general density theorem for Sobolev maps

Azevedo, Assis; Azevedo, Davide; Bessa, Mário; Torres, Maria

doi:10.1090/proc/15352

The closing lemma and the planar general density theorem for Sobolev maps
HTML articles powered by AMS MathViewer

by Assis Azevedo, Davide Azevedo, Mário Bessa and Maria Joana Torres

Proc. Amer. Math. Soc. 149 (2021), 1687-1696

DOI: https://doi.org/10.1090/proc/15352

Published electronically: February 12, 2021

PDF | Request permission

Abstract:

We prove that given a non-wandering point of a Sobolev-$(1,p)$ homeomorphism we can create closed trajectories by making arbitrarily small perturbations. As an application, in the planar case, we obtain that generically the closed trajectories are dense in the non-wandering set.

References

Marie-Claude Arnaud, $C^1$-generic billiard tables have a dense set of periodic points, Regul. Chaotic Dyn. 18 (2013), no. 6, 697–702. MR 3146587, DOI 10.1134/S1560354713060099
Masayuki Asaoka and Kei Irie, A $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces, Geom. Funct. Anal. 26 (2016), no. 5, 1245–1254. MR 3568031, DOI 10.1007/s00039-016-0386-3
Assis Azevedo, Davide Azevedo, Mário Bessa, and Maria Joana Torres, Sobolev homeomorphisms are dense in volume preserving automorphisms, J. Funct. Anal. 276 (2019), no. 10, 3261–3274. MR 3944293, DOI 10.1016/j.jfa.2018.10.008
Mário Bessa and Maria Joana Torres, The $C^0$ general density theorem for geodesic flows, C. R. Math. Acad. Sci. Paris 353 (2015), no. 6, 545–549 (English, with English and French summaries). MR 3348990, DOI 10.1016/j.crma.2015.03.012
Mário Bessa, Maria Joana Torres, and Paulo Varandas, On the periodic orbits, shadowing and strong transitivity of continuous flows, Nonlinear Anal. 175 (2018), 191–209. MR 3830728, DOI 10.1016/j.na.2018.06.002
E. M. Coven, J. Madden, and Z. Nitecki, A note on generic properties of continuous maps, Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), Progr. Math., vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 97–101. MR 670076
Edson de Faria, Peter Hazard, and Charles Tresser, Infinite entropy is generic in Hölder and Sobolev spaces, C. R. Math. Acad. Sci. Paris 355 (2017), no. 11, 1185–1189 (English, with English and French summaries). MR 3724883, DOI 10.1016/j.crma.2017.10.016
E. de Faria, P. Hazard, and C. Tresser, Genericity of infinite entropy for maps with low regularity, Preprint ArXiv 2017.
Mike Hurley, On proofs of the $C^0$ general density theorem, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1305–1309. MR 1307531, DOI 10.1090/S0002-9939-96-03184-X
Stanislav Hencl and Aldo Pratelli, Diffeomorphic approximation of $W^{1,1}$ planar Sobolev homeomorphisms, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 3, 597–656. MR 3776275, DOI 10.4171/JEMS/774
Michael-R. Herman, Différentiabilité optimale et contre-exemples à la fermeture en topologie $C^\infty$ des orbites récurrentes de flots hamiltoniens, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 1, 49–51 (French, with English summary). MR 1115947
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
Tadeusz Iwaniec, Leonid V. Kovalev, and Jani Onninen, Diffeomorphic approximation of Sobolev homeomorphisms, Arch. Ration. Mech. Anal. 201 (2011), no. 3, 1047–1067. MR 2824471, DOI 10.1007/s00205-011-0404-4
John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443, DOI 10.1007/978-1-4684-9339-9
Charles C. Pugh, The closing lemma, Amer. J. Math. 89 (1967), 956–1009. MR 226669, DOI 10.2307/2373413
Charles C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), 1010–1021. MR 226670, DOI 10.2307/2373414
Charles Pugh, Against the $C^{2}$ closing lemma, J. Differential Equations 17 (1975), 435–443. MR 368079, DOI 10.1016/0022-0396(75)90054-6
Charles C. Pugh and Clark Robinson, The $C^{1}$ closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems 3 (1983), no. 2, 261–313. MR 742228, DOI 10.1017/S0143385700001978
Ludovic Rifford, Closing geodesics in $C^1$ topology, J. Differential Geom. 91 (2012), no. 3, 361–381. MR 2981842
S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 97–116. MR 165537
Lan Wen, The $C^1$ closing lemma for nonsingular endomorphisms, Ergodic Theory Dynam. Systems 11 (1991), no. 2, 393–412. MR 1116648, DOI 10.1017/S0143385700006210