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I am having some trouble with the following:

Let $(X_1,\mathcal{F}_1,\mu_1,T_1)$ be a measure preserving dynamical system with $T_1:X_1\to X_1$ is ergodic.

Suppose $(X_2,\mathcal{F}_2,\mu_2,T_2)$ is another dynamical system and there exists some surjective function $\pi: X_2\to X_1$ such that $\pi\circ T_2=T_1\circ \pi$ and $\mu_1=\pi_\star\mu_2$ (i.e. the pushforward measure).

Can we deduce ergodicity of $T_2$ from that of $T_1$?

Remark: I can show invariance, I am only interested in ergodicity!

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No. Consider $X_2=X_1\coprod X_1$ (i.e. two disjoint copies of $X$) with the measure $\mu_2(A\coprod B)=\frac{1}{2}(\mu_1(A)+\mu_1(B))$ and $T_2$ be given by letting $T_1$ acting separately on each copy of $X_1$ and let $\pi:X_2\to X_1$ be given by the identity on each component. This is clearly a surjection from $X_2$ to $X_1$, and the image measure of $\mu_2$ under $\pi$ is clearly $\mu_1$, and $\pi$ intertwines the dynamics.

However, each copy of $X_1$is invariant in $X_2$, and has probability $\frac{1}{2}$ and hence, $T_2$ is not ergodic.

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