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

MathSciNet review:
743733

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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]**Rufus Bowen,*Equilibrium states and the ergodic theory of Anosov diffeomorphisms*, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR**0442989****[2]**Rufus Bowen and David Ruelle,*The ergodic theory of Axiom A flows*, Invent. Math.**29**(1975), no. 3, 181–202. MR**0380889****[3]**Rufus Bowen,*Periodic orbits for hyperbolic flows*, Amer. J. Math.**94**(1972), 1–30. MR**0298700****[4]**Rufus Bowen,*A horseshoe with positive measure*, Invent. Math.**29**(1975), no. 3, 203–204. MR**0380890****[5]**David Fried and Michael Shub,*Entropy, linearity and chain-recurrence*, Inst. Hautes Études Sci. Publ. Math.**50**(1979), 203–214. MR**556587****[6]**J. Palis,*On Morse-Smale dynamical systems*, Topology**8**(1968), 385–404. MR**0246316****[7]**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****[8]**Charles Pugh and Michael Shub,*The Ω-stability theorem for flows*, Invent. Math.**11**(1970), 150–158. MR**0287579****[9]**M. Shub,*Stability and genericity for diffeomorphisms*, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 493–514. MR**0331431****[10]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747–817. MR**0228014**, 10.1090/S0002-9904-1967-11798-1**[11]**Helena S. Wisniewski,*Rate of approach to minima and sinks: the 𝐶² Axiom A no cycle case*, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 789–813. MR**730300**, 10.1007/BFb0061447

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DOI:
http://dx.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