1
$\begingroup$

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.

$\endgroup$

1 Answer 1

2
$\begingroup$

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. 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 $X:=X_1 \times X_2$, $T:= T_1 \times T_2$, $F:=\hat{F} \times T_2$, $\mu:=\mu_1 \times \mu_2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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