Topological invariants for equivariant flows: Conley index and degree
About forty years have passed since Charles Conley defined the homotopy index. Thereby, he generalized the ideas that go back to the calculus of variations work of Marston Morse. Within this long time the Conley index has proved to be a valuable tool in nonlinear analysis and dynamical systems. A significant development of applied methods has been observed. Later, the index theory has evolved to cover such areas as discrete dynamical systems, or analysis of flows defined on locally noncompact spaces cf. LS -index. Using the Conley theory, one is interested in the behavior of the particular sets of solutions, called isolated invariant sets, of differential equations. The index of an isolated invariant set S is a homotopy type (or, in case of an LS -index, a stable homotopy type) of the quotient X/A of a certain pair, called the index pair. It will be denoted by h(S). Since the homotopy types cannot be lined up (like, for instance, the real numbers) and they are often very difficult to distinguish, they are fairly hard to work with. Thus, the cohomological index H∗(X/A) has been found to be more accessible to the applications. It it easier to compare this index with other algebraic topological characteristics of the dynamical systems. Probably the most important feature, among others, of the Conley index is the invariance with respect to small perturbations of the initial differential equation. A large collection of tools, called the homotopy invariants, has this special property. They include: the index of a zero of a vector field, topological degree, intersection number, Lefschetz number etc. Herein, we are focused on the topological degree and some of its extensions. The overall aim of this thesis is to study the relationship between the degree of a vector field and the Conley index of the induced flow. A large part of the thesis is devoted to the equivariant version of the Morse type inequalities (called equivariant Morse–Conley–Zehnder equation). The equivariant Morse inequalities have been used to compare the G-Conley index with the gradient equivariant degree. It was actually my primary intention. However, this Morse–Conley–Zehnder equation seems to be very useful in the critical point theory. Therefore, I decided to place some simple multiplicity results for critical orbits of invariant (with respect to the Lie group action) functions.
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