Isolated Invariant Sets and the Morse Index (Conference Board of the Mathematical Sciences Series No. 38)

This volume contains lectures from the Conference Board of Mathematical Sciences meeting held at the University of Colorado on May 31-June 4, 1976. The lectures consist of an expository discussion of basic results for topological flows and a somewhat more detailed discussion of isolated invariant sets and continuation. The construction of the index for isolated invariant sets is new and allows more general application than previous ones. Also, the index itself is endowed with more structure and the continuation theorem is modified to take this new structure into account. Some elementary applications are given, but the main emphasis is on the abstract theory. Title......Page 1 Contents......Page 3 Preface......Page 5 1. Isolated invariant sets and continuation......Page 7 2. An example......Page 8 3. The Morse index......Page 9 4. Sums and products of indices......Page 11 5. A consequence of the sum formula......Page 15 6. Gradient-like equations......Page 16 7. Attractors, repellers and Morse decompositions......Page 18 8. Chain recurrent and strongly gradient-like flows......Page 20 9. Some examples of bifurcation......Page 22 10. Concluding remarks......Page 26 2. Flows and the theorem of Wazewski......Page 28 3. The translation flow on the space of curves......Page 31 4. Limit sets, nonwandering sets and compact invariant sets......Page 33 5. Attractor-repeller pairs......Page 36 6. Chain recurrence......Page 40 7. Morse decompositions......Page 44 2. Definitions from homotopy theory......Page 47 3. Local flows and isolated invariant sets......Page 48 4. Index pairs......Page 50 5. The Morse index......Page 54 6. Computing the homotopy index......Page 57 7. Connections......Page 64 8. Concluding remarks......Page 67 1. The space of isolated invariant sets -......Page 68 2. Continuation of I(S)......Page 72 3. Continuation of connections and Morse decompositions.......Page 76 4. Cohomological aspects of structure......Page 78 5. Churchill's sequence, Montgomery's formulation and the Morse-Smale theorem......Page 80 6. Remarks about local flows defined by equations other than integral equations......Page 83 7. Flows with special properties......Page 86 8. Concluding remarks......Page 87 Bibliography......Page 89