What does it mean topological invariant?

Question

I have to show that unique ergodicity is a topological invariant.

What does it mean $\bf{topological\; invariant}$?

Thank you!

Answer 1

A topological invariant is first of all an application which take a topological space and return something algebraic (group, polynomial, matrix...).

We say that $\mathcal I$ is a topological invariant means that if $X$ and $Y$ are two topological spaces such that there exists an homeomorphism $h:X\to Y$ then

$$\mathcal I(X)=\mathcal I(Y).$$

Answer 2

In the context of dynamical systems a topological invariant is a function or a property of the dynamics that is invariant under topological conjugacy.

Most certainly, I agree with the other answer in the context of topology, but this is dynamics and there is more structure. This causes that the notion of topological invariant must address that structure as well. The term "topological invariant" for the notion that I give above in the context of dynamics is the canon in the area. Most of the time we consider only one space (and thus $X=Y$ in the other answer).

In what respect to unique ergodicity surely you wanted to say that you have a dynamics on a compact metrizable space, right? In this case, you simply need to use the topological conjugacy to show that for the new dynamics you also have uniform convergence of the Birkhoff averages. Otherwise, you have a problem (in general your question would make no sense).