On Properties of Expansive Group Actions

Abstract

We study several properties of expansive group actions on metric spaces and obtain relation between expansivity for subgroup and group actions. Through counter examples necessity of hypothesis are justified. We also study expansivity of covering maps. Further, we define orbit expansivity for group actions on topological spaces and use it to characterize expansive action.

Chaotic behaviour of countable products of homeomorphism groups

“…Definition 5.1. The metric d on X = i∈N X i given by the formula (5), is called by the direct product of metrics d i . The metric space (X, d) is called the direct product of countable family of metric spaces (X i , d i ) and it is denoted by (X, d) = i∈N (X i , d i ).…”

“…Example 6.3. Recall that a homeomorphism group G of a metric space (X, d) is said to be expansive on X, if there exists a constant c > 0 such that for every x = y in X, there is g ∈ G satisfying d(g.x, g.y) > c. Such a constant c is called an expansivity constant of this group [5]. Every expansive group G of homeomorphisms of a metric space (X, d) is sensitive to initial conditions, and the role of δ in Definition 6.2 plays the expansivity constant c. Proposition 6.4.…”

“…We use notations from Section 5. In the case when J is a finite subset of N, the metric d of the product i∈J (X i , d i ) is defined by the formula (5), in which i ∈ J. The sensitivity of the group G = i∈J G i is not changed if the metric d will be defined by the formula…”

<img src=image/13491345_01.gif>-action Induced by Shift Map on 1-Step Shift of Finite Type over Two Symbols and k-type Transitive

“…There are many finding results which focus on group actions and also some other special kind of group actions. For instance, Barzanouni et al [1] had studied group actions on metric space with expansive properties. They found several relations of expansivity in various cases such as between subgroup actions and covering maps.…”

Measurable Version of Spectral Decomposition Theorem for a Z2-Action

“…It is natural to consider other notions of expansiveness, such as N-expansive [23], G-expansive [24] and pointwise expansive [25]. Some studies have already defined expansiveness for flows [26], group actions [27], Z d -actions [28], etc. Morales and Sirvent were able to develop and introduce a notion of expansivity by using the theory of measures.…”