Stability of skew dynamical systems (Q1817260)

Let \(M\) be a compact differential manifold and \(T\) be a compact metric space. The paper considers a dynamical system induced by a homeomorphism \(f:M \times T \to M \times T\) of the form \(f(x,t) = (\Psi (x,t),\;V(t))\), where \(\Psi (x,t)\) is a diffeomorphism for fixed \(t\), and \(V: T \to T\) is an almost periodic homeomorphism. Here `almost periodic' is in the sense of Bohr. Such a system is called a smooth skew dynamical system over the base \(V: T \to T\). The authors generalize the concepts of stable and unstable manifolds, hyperbolic structure, shadowing, basic set, etc. The main result is a generalization of Smale's \(\Omega\)-stability theory, which asserts that an ordinary dynamical system is \(\Omega\)-stable if it satisfies Axiom A and has the no-cycle property.