why topological conjugacy does not preserve ergodicity?

Question

I want to know why topological conjugacy does not transfer ergodicity from one dynamical system to another?and if we change the maps from continous to C1-differentiable will the problem solve or not?

Answer 1

As far as I am aware, topological conjugacy does transfer ergodicity. That is, if the map $\tau: X \to X$ preserves the Borel probability measure $\mu$ and is ergodic (for that measure), and $\phi: X \to Y$ is a homeomorphism, then $\phi \circ \tau \circ \phi^{-1}: Y \to Y$ preserves the Borel probability measure $\mu \circ \phi^{-1}$ and is ergodic (for that measure).

What topological conjugacy doesn't preserve is absolute continuity of the invariant measure with respect to, say, Lebesgue measure.

Answer 2

It depends what you mean. The problem of topological conjugacy is that it can be not absolutely continuous (but if you ask for $C^1$, it is), which implies that it can map a set of positive measure to a set of zero measure or set of zero measure to a set of positive measure. So if you in a bad situation, you have one measure with a nice SRB (ergodic with Lebsgue as a reference measure) measure, for example, after applying a homeo you can get a measure that would be ergodic (as R. Israel said), but will not be SRB.