Stability Properties of One-Parameter Flows

Abstract

This thesis is in two parts. The first part consists of chapters 1-4 and the second part is chapter 5. In the first part we consider the idea of approximating pieces of orbits by a single orbit. There are many examples of such properties (approximation property (A.P.) see chapter one, definitions 1.2, 1.4), (Specification property (S.P.),[24]), (pseudo orbit tracing property (P.O.T.P.) [25] and definition 4.1). In chapter one, we show that (A.P.) for a homeomorphism (flow) is equivalent to topological transitivity and density of periodic points and how this property (A.P.) is invariant under topological conjugacy. In theorem 1 we prove that an expansive homeomorphism which is topologically mixing and has P.O.T.P. also has the S.P. In chapter two, theorem 2, we prove that the P.O.T.P. for flows is invariant under topological conjugacy with preserved orientation (velocity changes). Also we prove in theorem 3 that the suspension flow [2] for a homeomorphism T:X → X has P.O.T.P. if and only if T has the P.O.T.P. In chapter three, we prove that an expansive flow which has the P.O.T.P. is topologically stable (Theorem 4). In chapter four, the last theorem in this part is that every flow without fixed points on a compact manifold M which is topologically stable has the P.O.T.P. Then some important corollaries are deduced. In the second part of this thesis … [please see full thesis for rest of abstract]