The $C^1$ closing lemma for nonsingular endomorphisms equivariant under free actions of finite groups
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- by Xiaofeng Wang and Duo Wang PDF
- Trans. Amer. Math. Soc. 351 (1999), 4173-4182 Request permission
Abstract:
In this paper a closing lemma for $C^1$ nonsingular endomorphisms equivariant under free actions of finite-groups is proved. Hence a recurrent trajectory, as well as all of its symmetric conjugates, of a $C^1$ nonsingular endomorphism equivariant under a free action of a finite group can be closed up simultaneously by an arbitrarily small $C^1$ equivariant perturbation.References
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
- S. T. Liao, An extension of the $C^1$ closing lemma, Beijing Daxue Xuebao 3 (1979), 1–41 (in Chinese).
- Jie Hua Mai, A simpler proof of $C^1$ closing lemma, Sci. Sinica Ser. A 29 (1986), no. 10, 1020–1031. MR 877286
- Jie Hua Mai, A simpler proof of the extended $C^1$ closing lemma, Chinese Sci. Bull. 34 (1989), no. 3, 180–184. MR 1001816
- Ricardo Mañé, A proof of the $C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math. 66 (1988), 161–210. MR 932138
- 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 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
- 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
- Lan Wen, The $C^1$ closing lemma for endomorphisms with finitely many singularities, Proc. Amer. Math. Soc. 114 (1992), no. 1, 217–223. MR 1087474, DOI 10.1090/S0002-9939-1992-1087474-6
- Lan Wen, On the $C^1$ stability conjecture for flows, J. Differential Equations 129 (1996), no. 2, 334–357. MR 1404387, DOI 10.1006/jdeq.1996.0121
Additional Information
- Xiaofeng Wang
- Affiliation: Department of Applied Mathematics, Tsinghua University, Beijing 100084, P.R. China
- Email: xfwang@math.tsinghua.edu.cn
- Duo Wang
- Affiliation: School of Mathematical Science, Peking University, Beijing 1000871, P.R. China
- Email: dwang@sxx0.math.pku.edu.cn
- Received by editor(s): February 21, 1997
- Published electronically: March 18, 1999
- Additional Notes: This work is supported by NNSF of China.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 4173-4182
- MSC (1991): Primary 58F10, 58F20, 58F22, 58F35
- DOI: https://doi.org/10.1090/S0002-9947-99-02199-6
- MathSciNet review: 1473457