Rate of approach to minima and sinks--the Morse-Smale case

Author:
Helena S. Wisniewski

Journal:
Trans. Amer. Math. Soc. **284** (1984), 567-581

MSC:
Primary 58F09

DOI:
https://doi.org/10.1090/S0002-9947-1984-0743733-5

MathSciNet review:
743733

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The dynamical systems herein are Morse-Smale diffeomorphisms and flows on compact manifolds. We show the asymptotic rate of approach of orbits to the sinks of the systems to be bounded by an expression of the form , where may be any number smaller than . Here the minimum is taken over all nonsink in the nonwandering set of , and is the period of . We extend our theorems to the entire manifold, so that there is no restriction on the location of the initial points of trajectories.

**[1]**R. Bowen,*Equilibrium states and the ergodic theory of Anosov diffeomorphisms*, Lecture Notes in Math, vol. 470, Springer-Verlag, Berlin and New York, 1975. MR**0442989 (56:1364)****[2]**R. Bowen and D. Ruelle,*The ergodic theory of Axiom**flows*, Invent. Math.**29**(1975), 181-202. MR**0380889 (52:1786)****[3]**R. Bowen,*Periodic orbits of hyperbolic flows*, Amer. J. Math.**94**(1972), 1-37. MR**0298700 (45:7749)****[4]**-,*A horseshoe with positive measure*, Invent. Math.**29**(1975), 203-204. MR**0380890 (52:1787)****[5]**D. Fried and Michael Shub,*Entropy, linearity, and chain recurrence*, Extrait des Publications Mathematiques, No. 50, 1978, pp. 203-214. MR**556587 (81a:58033)****[6]**J. Palis,*On Morse-Smale dynamical systems*, Topology, Vol. 8, Pergamon Press, New York and Oxford, 1969, 385-405. MR**0246316 (39:7620)****[7]**J. Palis and S. Smale,*Structural stability theorems*, Proc. Sympos. Pure Math, Vol. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 223-232. MR**0267603 (42:2505)****[8]**C. Pugh and M. Shub,*The*-*stability theorem for flows*, Invent. Math.**11**(1970), 150-158. MR**0287579 (44:4782)****[9]**M. Shub,*Stability and genericity for diffeomorphisms*, Dynamical Systems (M. Peixoto, ed.), Academic Press, New York, 1973, pp. 493-514. MR**0331431 (48:9764)****[10]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**71**(1967), 747-814. MR**0228014 (37:3598)****[11]**H. Wisniewski,*Rate of approach to minima and sinks--the**Axiom**no cycles case*, Geometric Dynamics (Proc. Internat. Sympos. in Dynamical Systems, Rio de Janiero, Brazil, 1981), Lecture Notes in Math, vol. 1007, Springer-Verlag, Berlin and New York. MR**730300 (85b:58102)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F09

Retrieve articles in all journals with MSC: 58F09

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1984-0743733-5

Keywords:
Dynamical systems,
diffeomorphism,
Morse-Smale systems,
Axiom systems,
no-cycles,
transversality,
filtration,
hyperbolic invariant set,
basic set attractor,
flow,
unstable and stable manifold

Article copyright:
© Copyright 1984
American Mathematical Society