Bundle-Like Pre-Image Entropy

In this thesis we systematically study a new kind entropy-like invariants in the case of compact metric space and probability space. We organize the thesis as follows:In chapter one, the historic background and some results of the entropies and pressure are introduced briefly.In chapter two, we define and study bundle-like pre-image entropy which is a generalization to the standard notion of topological entropy and Cheng-Newhouse pre-image entropy. We obtain analogs of many known results for topological entropy, such as product rules, power rules and topological invariance. Last, we show how to calculate bundle-like preimage entropy under the assumption of forward expansiveness.In chapter three, we introduce a notion of pre-image conditional measure-theoretic entropy for an T-invariant sub-a-algebra of a probability space (X, B,μ), namely h_pre,μ(T|A). We show that h_pre,μ(T|A) satisfies power and product rules, the mapμ→h_pre,μ(T|A) is affine. Last, we give the ergodic decomposition of pre-image conditional measure-theoretic entropy.In chapter four, we obtain a variational principle for bundle-like pre-image entropy and pre-image conditional measure-theoretic(or metric) entropy. More precisely, for a given dynamical system (X, T) (where X is a compact metirc space and T is a continuous map from X into itself) and a Borel partitionξof X with T^-1ξ≤ξ, we prove a variational principle for bundle-like pre-image entropy h_b(T|ξ),where |ξ| denote the sub-a-algebra generated byξand M (X, T) denote the set of invariant measures of T.