Pugh, Charles C. The \(C^{1+\alpha}\) hypothesis in Pesin theory. (English) Zbl 0542.58027 Publ. Math., Inst. Hautes Étud. Sci. 59, 143-161 (1984). The stable manifold theory developed by Pesin and others contains a hypothesis that the given dynamics be of class \(C^{1+\alpha}\) for some \(\alpha>0\), i.e. the first derivatives must obey \(\alpha\)-Hölder conditions. This paper shows that \(C^ 1\) alone, i.e. \(\alpha =0\), is insufficient as follows. Theorem: There exists a \(C^ 1\) diffeomorphism of a 4-manifold having an asymptotically hyperbolic orbit O(p) such that \(W^ s(p)\) is not an injectively immersed manifold tangent to \(E^ s_ p\). Reviewer: G.Ikegami Cited in 1 ReviewCited in 19 Documents MSC: 37D99 Dynamical systems with hyperbolic behavior 58C25 Differentiable maps on manifolds Keywords:stable manifold; diffeomorphism PDF BibTeX XML Cite \textit{C. C. Pugh}, Publ. Math., Inst. Hautes Étud. Sci. 59, 143--161 (1984; Zbl 0542.58027) Full Text: DOI Numdam EuDML References: [1] A. Fathi, M. Herman andJ. C. Yoccoz,A proof of Pesin’s Stable Manifold Theorem, preprint of Université de Paris-Sud, Orsay, France. · Zbl 0532.58012 [2] E. A. Gonzales Velasco, Generic Properties of Polynomial Vector Fields at Infinity,Trans. AMS,143 (69), 201–222. · Zbl 0187.34401 [3] M. Hirsch, C. Pugh andM. Shub, Invariant Manifolds,Springer Lecture Notes,583, 1977. [4] A. Katok, Lyapunov Exponents, Entropy and Periodic Orbits for Diffeomorphisms,Publ. Math. IHES,51 (1980), 137–173. · Zbl 0445.58015 [5] R. Mañé Ramirez,Introducão à Teoria Ergódica, IMPA, 1979. [6] V. I. Oseledec, Multiplicative Ergodic Theorem, Lyapunov Characteristic Exponents for Dynamical Systems,Trans. Moscow Math. Soc.,19 (1968), 197–231. · Zbl 0236.93034 [7] Y. B. Pesin, Families of Invariant Manifolds Corresponding to Nonzero Characteristic Exponents,Math. USSR Izvestija,10 (1976), 1261–1305. · Zbl 0383.58012 · doi:10.1070/IM1976v010n06ABEH001835 [8] Y. B. Pesin, Characteristic Lyapunov Exponents and Smooth Ergodic Theory,Russian Math. Surveys,82 (1977), 4, 55–114. · Zbl 0383.58011 · doi:10.1070/RM1977v032n04ABEH001639 [9] D. Ruelle, Ergodic Theory of Differentiable Dynamical Systems,Publ. Math. IHES,50 (1979), 27–58. · Zbl 0426.58014 [10] S. Sternberg, Local C n Transformations of the Real Line,Duke Mathematical Journal,24 (1957), 97–102. · Zbl 0077.06201 · doi:10.1215/S0012-7094-57-02415-8 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.