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Characterization of MorseSmale isotopy classes on surfaces

Published online by Cambridge University Press:  19 September 2008

Luiz Fernando  and
C. Da Rocha
Luiz Fernando
Affiliation:
Instituto de Matemática da UFRGS, Porto Alegre, RS 90.000, Brazil
C. Da Rocha
Affiliation:
Instituto de Matemática da UFRGS, Porto Alegre, RS 90.000, Brazil
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Abstract

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In this paper we use Thurston's work on the dynamics of diffeomorphisms on surfaces to show that a diffeomorphism ƒ on a surface is isotopic to a Morse- Smale one if and only if the growth rate of the length of the words representing elements of the fundamental group under iteration by ƒ is one. Morse-Smale isotopy classes are also shown to be the same as Nielsen's algebraically finite type.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 1 , March 1985 , pp. 107 - 122
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[0]Bers, L.. An extremal problem for quasiconformal mappings and a theorem by Thurston. Acta Math. 141 (1978), 73–98.CrossRefGoogle Scholar
[1]Bishop, R. L. & Crittenden, R. J.. Geometry of Manifolds. Academic Press, 1964.Google Scholar
[2]Bowen, R.. Entropy and the Fundamental Group. Lecture Notes in Mathematics 668, pp 21–29. Springer-Verlag: Berlin-Heidelberg-New York, 1975.Google Scholar
[3]Manning, A.. Topological entropy and the first homology group. In Proc. Symp. Dyn. Syst. Warwick 1973/1974, pp 185–190.Google Scholar
[4]Milnor, J.. A note on curvature and the fundamental group. J. Diff. Geom. 2 (1968), 1–7.Google Scholar
[5]Nielsen, J.. Surface transformations of algebraically finite type. Danske. Vid. Selk. Math-Phys. Medd. 21, no. 2, 88 pp (1944).Google Scholar
[6]Shub, M.. Morse-Smale diffeomorphisms are unipotent on homology. Dynamical Systems, Proc. Symp. Salvador (1971) pp 489–491.Google Scholar
[7]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[8]Thurston, W. P.. On the geometry and dynamics of diffeomorphisms of surfaces. Preprint. (See also Astérique 66–67: Travaux de Thurston surles surfaces; Fathi, A., Laudenbach, F., Poénaru, V. (Editors).)Google Scholar