Why, in the study of topological dynamical systems, it is required that the metric space is compact or locally compact?

Question

I am studying the book Dynamical Systems: An Introduction by Barreira and Valls, and at the beginning of Chapter 3 it is mentioned that for simplicity it will always be supposed in the book that $X$ is a locally compact metric space. I didn't understand why the space must be compact. I appreciate any help.

Answer 1

Barreira and Valls state they also consider the space to be second countable. As they state, and as you state as they state, local compactness, separability, or even that the space is endowed with a metric, is to simplify the exposition. There is no a priori reason why one could not consider continuous/uniform self-maps of a topological/uniform space and still consider the study part of topological dynamics.

As an example, which is somewhat related to another question of yours (Relationship between hyperbolicity of linear flow and eigenvalue with norm $1$), one can consider iterates of bounded linear automorphisms of Banach spaces (as in Space of linear, continuous, hyperbolic functions is open, dense in the set of invertible functions ). If the dimension of the Banach space is not finite, then such a space is not locally compact (see e.g. Locally-compact function spaces?).