In this dissertation, we study the complicated dynamics of two classes of systems: Anosov systems driven by an external force and partially hyperbolic systems driven by an external force. For smooth Anosov systems driven by an external force, we first study the random specification property, which is on the approximation of an N−spaced arbitrary long finite random orbit segments within given precision by a random periodic point. We prove that if such system is topological mixing on fibers, then it has the random specification property. Furthermore, we prove that the homeomorphism induced by such a system on the space of random probability measures also has the specification property. We note that the random specification property implies the positivity of topological fiber entropy. Secondly, we show that if the system is topological mixing on fibers, then its past and future random correlation for Hölder observable functions decay exponentially with respect to the system and the unique random SRB measure. For smooth partially hyperbolic systems driven by an external force, we prove the existence of the random Gibbs u−state, which has absolutely continuous conditional measure on the strong unstable manifolds.
College and Department
Physical and Mathematical Sciences; Mathematics
BYU ScholarsArchive Citation
Liu, Xue, "Dynamics of Systems Driven by an External Force" (2021). Theses and Dissertations. 9425.
random dynamical systems, random specification, Bowen's specification property, exponential decay of random correlation, absolute continuity, random SRB measure, Birkhoff cone, random Gibbs uâˆ’state