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 $X$ with strictly positive Sinai entropy $h\_{\mu}(T).$ 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.



Non-convergence of ergodic measures with positive entropy
Let $T\_1, \hat{F} \colon X\_1 \to X\_1$ be commuting continuous transformations of a compact metric space and $\mu\_1$ a Borel probability measure on $X\_1$ which is invariant and ergodic with respect to $T\_1$, and such that $\frac{1}{N}\sum\_{k=1}^N {\hat{F}}\_\*^k\mu\_1$ does not converge, as in [your previous question](https://mathoverflow.net/questions/168812). Let $T\_2 \colon X\_2 \to X\_2$ be a continuous transformation of a compact metric space equipped with a $T\_2$-invariant weak-mixing Borel probability measure $\mu\_2$ such that $h(T\_2,\mu\_2)>0$. Define $X:=X\_1 \times X\_2$, $T:= T\_1 \times T\_2$, $F:=\hat{F} \times T\_2$, $\mu:=\mu\_1 \times \mu\_2$. 


