Geodesic
Lusternik--Schnirelman Theory and Dynamics. II
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 252-266.

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We show how the methods of homotopy theory can be used in dynamics to study the topology of a chain recurrent set. More specifically, we introduce new homotopy invariants $\mathrm {cat}^1(X,\xi)$ and $\mathrm {cat}^1_{\mathrm s}(X,\xi)$ that depend on a finite polyhedron $X$ and a real cohomology class $\xi \in H^1(X;\mathbb R)$ and are modifications of the invariants introduced earlier by the first author. We prove that, under certain conditions, $\mathrm {cat}_{\mathrm s}^1(X,\xi)$ provides a lower bound for the Lusternik–Schnirelman category of the chain recurrent set $R_\xi$ of a given flow. The approach of the present paper applies to a wider class of flows compared with the earlier approach; in particular, it allows one to avoid certain difficulties when checking assumptions. 
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     title = {Lusternik--Schnirelman {Theory} and {Dynamics.} {II}},
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M. Farber; T. Kappeler. Lusternik--Schnirelman Theory and Dynamics. II. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 252-266. http://geodesic.mathdoc.fr/item/TM_2004_247_a18/

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