Axiom A flows with a transverse torus

Morales, C.

doi:10.1090/S0002-9947-02-03127-6

Axiom A flows with a transverse torus
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by C. A. Morales PDF

Trans. Amer. Math. Soc. 355 (2003), 735-745 Request permission

Abstract:

Let $X$ be an Axiom A flow with a transverse torus $T$ exhibiting a unique orbit $O$ that does not intersect $T$. Suppose that there is no null-homotopic closed curve in $T$ contained in either the stable or unstable set of $O$. Then we show that $X$ has either an attracting periodic orbit or a repelling periodic orbit or is transitive. In particular, an Anosov flow with a transverse torus is transitive if it has a unique periodic orbit that does not intersect the torus.

References

Joan S. Birman and R. F. Williams, Knotted periodic orbits in dynamical system. II. Knot holders for fibered knots, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 1–60. MR 718132, DOI 10.1090/conm/020/718132
Christian Bonatti and Rémi Langevin, Un exemple de flot d’Anosov transitif transverse à un tore et non conjugué à une suspension, Ergodic Theory Dynam. Systems 14 (1994), no. 4, 633–643 (French, with English summary). MR 1304136, DOI 10.1017/S0143385700008099
Marco Brunella, Separating the basic sets of a nontransitive Anosov flow, Bull. London Math. Soc. 25 (1993), no. 5, 487–490. MR 1233413, DOI 10.1112/blms/25.5.487
Marco Brunella, On the discrete Godbillon-Vey invariant and Dehn surgery on geodesic flows, Ann. Fac. Sci. Toulouse Math. (6) 3 (1994), no. 3, 335–344 (English, with English and French summaries). MR 1313353, DOI 10.5802/afst.783
Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541, DOI 10.1007/978-1-4612-5703-5
John Franks and Bob Williams, Anomalous Anosov flows, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 158–174. MR 591182
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
Michael Handel and William P. Thurston, Anosov flows on new three manifolds, Invent. Math. 59 (1980), no. 2, 95–103. MR 577356, DOI 10.1007/BF01390039
Gilbert Hector and Ulrich Hirsch, Introduction to the geometry of foliations. Part B. Foliations of codimension one, Aspects of Mathematics, E3, Friedr. Vieweg & Sohn, Braunschweig, 1983. MR 726931, DOI 10.1007/978-3-322-85619-7
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173, DOI 10.1007/BFb0092042
R. V. Plykin, Sources and sinks of $\textrm {A}$-diffeomorphisms of surfaces, Mat. Sb. (N.S.) 94(136) (1974), 243–264, 336 (Russian). MR 0356137
Clark Robinson, Dynamical systems, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. Stability, symbolic dynamics, and chaos. MR 1396532
Michael Shub, Stabilité globale des systèmes dynamiques, Astérisque, vol. 56, Société Mathématique de France, Paris, 1978 (French). With an English preface and summary. MR 513592