Example of non-convergence of iteration of measures

Question

Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X.$ Let $F:X\to X$ be a continuos transformation that commutes with $T.$

Define $F_{*}\mu(A)=\mu(F^{-1}A)$ for every Borel set $A.$

I am looking for an example in ergodic theory in which

$\frac{1}{N}\sum_{k=1}^NF_{*}^k\mu$ does not converge in the weak* topology as $N\to \infty.$

Pd: It is trivial to find examples in which $F_{*}^N\mu$ does not converge but $\frac{1}{N}\sum_{k=1}^NF_{*}^k\mu$ does.

Answer 1

Take $T$ to be the identity and $\mu$ the Dirac measure on a point $x \in X$. Obviously $\mu$ is $T$-invariant and $F$ commutes with $T$. Using well-known properties of the weak-* topology, the sequence of measures $\frac{1}{N}\sum_{k=1}^N F_*^k\mu$ converges if and only if for every continuous $f \colon X \to \mathbb{R}$, the sequence $\int f\,d\left(\frac{1}{N}\sum_{k=1}^NF_*^k\mu\right)=\frac{1}{N}\int \sum_{k=1}^N (f \circ F^k)d\mu = \frac{1}{N}\sum_{k=1}^N f(F^kx)$ converges. It is a simple exercise to construct examples where this convergence does not hold, for example if $F \colon X \to X$ is the 2-shift.