A New Topological Entropy: I Bear The Topological Entropy

The purpose of this paper is to introduce an entropy as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability not necessarily required). This is achieved through the consideration of the so-called co-compact covers of a topological space. The advantages of this new entropy include: 1) it does not require the space to be compact, and thus generalizes Adler, Konheim and McAndrew's topological entropy of continuous mappings on compact dynamical systems, and 2) it is an invariant of topological conjugation, compared to Bowen's entropy that is metric-dependent. Fundamental properties of this new entropy are investigated, e.g., the entropy of a subsystem does not exceed that of the whole system and topologically conjugated systems hold a same entropy. With this new definition, the entropy of the linear system (R, f) defined by f(x) = 2x is zero, which however is at least log 2 according to Bowen's definition of entropy. More general, this new entropy gives a lower bound of Bowen's entropies, which is proved through a generalized Lebesgue Covering Theorem for co-compact covers of non-compact metric spaces.Instead of using all open covers of the space to define entropy, we consider the open covers consisting of the so-called co-compact open sets (open sets whose complements are compact).Various definitions of entropy and historical notes are mentioned previously in this section, chapter 2 introduces the concept of co-compact open covers of a space and explores the topological properties of such covers and the new topological entropy defined based on co-compact covers of the space, what's more, in section 2.4 further explores the properties of this new entropy and compares it with Adler, Konheim and McAndrew's topological entropy for compact spaces. Sections 3 and 4 investigate the relation between this new entropy and Bowen's entropy. More precisely, Section 3 compares the new entropy with that given by Bowen for systems defined on metric spaces. Because the spaces under consideration include non-compact metric spaces, the traditional Lebesgue Covering Theorem does not apply. Thus, one work is to generalize this theorem to co-compact open covers of non-compact metric spaces. Based on the generalized Lebesgue covering theorem, we show that the new entropy is a lower bound for Bowen's entropies. In Section 4, a linear dynamical system is studied. For this simple system, its new entropy is 0 which is appropriate, but Bowen's entropy is positive.