# Specification for Group Actions on Uniform Spaces

## TL;DR

This paper extends the concept of specification to group actions on uniform spaces, establishing conditions under which such actions exhibit positive entropy and chaos, thus broadening the understanding of dynamical complexity in these systems.

## Contribution

It introduces a new notion of specification point for group actions on uniform spaces and proves results linking specification points to entropy and chaos.

## Key findings

- Group actions with two distinct specification points have positive entropy.
- Periodic specification implies Devaney chaos for certain group actions.
- Extension of specification concepts to uniform spaces broadens dynamical systems theory.

## Abstract

We extend specification and periodic specification to finitely generated group actions on uniform spaces using a concept of specification point. We prove that certain group actions having two distinct specification points have positive entropy. We further prove that if a group containing an infinite order element acts on an infinite Hausdorff uniform space and the action possesses periodic specification, then it is Devaney chaotic.