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!


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.