Expansiveness, generators and Lyapunov exponents for random bundle transformations

We generalize Fathi's results by showing that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a compatible hyperbolic metric. Moreover, we prove that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a generator in detail. Furthermore, we show that a fiber expansive homeomorphism has finite fiber topological entropy. Finally, we show that fiber Lyapunov exponents for a fiber expansive system are different from zero, indicating that the system presents a chaotic system. Meanwhile, we also prove that negative fiber Lyapunov exponents for compact invariant sets of a dynamical system imply that the compact set is a fiber attractor.