A proof of stability conjecture for three-dimensional flows

Author:
Sen Hu

Journal:
Trans. Amer. Math. Soc. **342** (1994), 753-772

MSC:
Primary 58F10; Secondary 58F15

DOI:
https://doi.org/10.1090/S0002-9947-1994-1172297-6

MathSciNet review:
1172297

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a proof of the stability conjecture for three-dimensional flows, i.e., prove that there exists a hyperbolic structure over the set for the structurally stable three-dimensional flows. Mañé's proof for the discrete case motivates our proof and we find his perturbation techniques crucial. In proving this conjecture we have overcome several new difficulties, e.g., the change of period after perturbation, the ergodic closing lemma for flows, the existence of dominated splitting over where is the set of singularities for the flow, the discontinuity of the contracting rate function on singularities, etc. Based on these we finally succeed in separating the singularities from the other periodic orbits for the structurally stable systems, i.e., we create unstable saddle connections if there are accumulations of periodic orbits on the singularities.

**[1]**Nobuo Aoki,*The set of Axiom A diffeomorphisms with no cycle*, Dynamical systems and related topics (Nagoya, 1990) Adv. Ser. Dynam. Systems, vol. 9, World Sci. Publ., River Edge, NJ, 1991, pp. 20–35. MR**1164874****[2]**C. I. Doering,*Persistently transitive vector fields on three-dimensional manifolds*, Dynamical systems and bifurcation theory (Rio de Janeiro, 1985) Pitman Res. Notes Math. Ser., vol. 160, Longman Sci. Tech., Harlow, 1987, pp. 59–89. MR**907891****[3]**John Franks,*Necessary conditions for stability of diffeomorphisms*, Trans. Amer. Math. Soc.**158**(1971), 301–308. MR**0283812**, https://doi.org/10.1090/S0002-9947-1971-0283812-3**[4]**J. E. Marsden and M. McCracken,*The Hopf bifurcation and its applications*, Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale; Applied Mathematical Sciences, Vol. 19. MR**0494309****[5]**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****[6]**Shan Tao Liao,*A basic property of a certain class of differential systems*, Acta Math. Sinica**22**(1979), no. 3, 316–343 (Chinese, with English summary). MR**549216****[7]**-,*An extension of the**closing lemma*, Acta Sci. Natur. Univ. Pekinensis**2**(1979), 1-41.**[8]**Shan Tao Liao,*Obstruction sets. II*, Beijing Daxue Xuebao**2**(1981), 1–36 (Chinese, with English summary). MR**646519****[9]**Shan Tao Liao,*On the stability conjecture*, Chinese Ann. Math.**1**(1980), no. 1, 9–30 (English, with Chinese summary). MR**591229****[10]**Shan Tao Liao,*Standard systems of differential equations and obstruction sets with applications to structural stability problems*, Proceedings of the 1983 Beijing symposium on differential geometry and differential equations, Sci. Press Beijing, Beijing, 1986, pp. 65–97. MR**880871****[11]**Ricardo Mañé,*Persistent manifolds are normally hyperbolic*, Trans. Amer. Math. Soc.**246**(1978), 261–283. MR**515539**, https://doi.org/10.1090/S0002-9947-1978-0515539-0**[12]**Ricardo Mañé,*An ergodic closing lemma*, Ann. of Math. (2)**116**(1982), no. 3, 503–540. MR**678479**, https://doi.org/10.2307/2007021**[13]**Ricardo Mañé,*On the creation of homoclinic points*, Inst. Hautes Études Sci. Publ. Math.**66**(1988), 139–159. MR**932137****[14]**-,*A proof of the**stability conjecture*, Inst. Hautes Etudes Sci. Publ. Math.**66**(1987), 161-210.**[15]**V. V. Nemytskii and V. V. Stepanov,*Qualitative theory of differential equations*, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. MR**0121520****[16]**J. Palis,*On the 𝐶¹ Ω-stability conjecture*, Inst. Hautes Études Sci. Publ. Math.**66**(1988), 211–215. MR**932139****[17]**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****[18]**J. Palis and S. Smale,*Structural stability theorems*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 223–231. MR**0267603****[19]**V. A. Pliss,*A hypothesis due to Smale*, Differential Equations**8**(1972), 203-214.**[20]**Charles C. Pugh,*The closing lemma*, Amer. J. Math.**89**(1967), 956–1009. MR**0226669**, https://doi.org/10.2307/2373413**[21]**J. W. Robbin,*A structural stability theorem*, Ann. of Math. (2)**94**(1971), 447–493. MR**0287580**, https://doi.org/10.2307/1970766**[22]**Clark Robinson,*Structural stability of 𝐶¹ diffeomorphisms*, J. Differential Equations**22**(1976), no. 1, 28–73. MR**0474411**, https://doi.org/10.1016/0022-0396(76)90004-8**[23]**Atsuro Sannami,*The stability theorems for discrete dynamical systems on two-dimensional manifolds*, Nagoya Math. J.**90**(1983), 1–55. MR**702250****[24]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747–817. MR**0228014**, https://doi.org/10.1090/S0002-9904-1967-11798-1

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F10,
58F15

Retrieve articles in all journals with MSC: 58F10, 58F15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1172297-6

Article copyright:
© Copyright 1994
American Mathematical Society