Poincaré-Hopf inequalities for periodic orbits

The interplay between the dynamics of a nonsingular Morse-Smale flow on a smooth, closed, n -dimensional manifold, M , and the topology of M , was exhibited in Franks (Comment Math Helv 53(2):279–294, 1978 ), Smale (Bull Am Math Soc 66:43–49, 1960 ), by means of a collection of inequalities, which w...

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Bibliographic Details
Published inGeometriae dedicata Vol. 160; no. 1; pp. 147 - 167
Main Author Bertolim, M. A.
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.10.2012
Subjects
Algebraic Geometry
Convex and Discrete Geometry
Differential Geometry
Hyperbolic Geometry
Mathematics
Mathematics and Statistics
Original Paper
Projective Geometry
Topology
Morse-Smale flows
Conley index theory
37B30
37D15
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Summary:The interplay between the dynamics of a nonsingular Morse-Smale flow on a smooth, closed, n -dimensional manifold, M , and the topology of M , was exhibited in Franks (Comment Math Helv 53(2):279–294, 1978 ), Smale (Bull Am Math Soc 66:43–49, 1960 ), by means of a collection of inequalities, which we refer to as Morse-Smale inequalities. These inequalities relate the number of closed orbits of each index to the Betti numbers of M . These well-known inequalities provide the necessary conditions for a given dynamical data in the form of a specified number of closed orbits of a given index to be realized as a nonsingular Morse-Smale flow on M . In this article we provide two inequalities, hereby referred to as Poincaré-Hopf inequalities for periodic orbits, which imposes constraints on the dynamics of periodic orbits without reference to the Betti numbers of the manifold M . The main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse-Smale inequalities to hold.
ISSN:0046-5755
1572-9168
DOI:10.1007/s10711-011-9673-1