# Ergodicity of semiconjugacy

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!

## 1 Answer

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.