Let $(X,B,m)$ be the particular probability space where $X$ is the circle in the plane with center at the origin and radius 1, $B$ is the collection of borel sets, $m$ is the lebesgue measure. Let $T:X\rightarrow X$ denote rotation by $\frac{2\pi }{3}$ radians. We know that $T$ is measure preserving. Let $A$ be the set of points $(x,y)\in X$ with $x>0$ and $y>0$ and let f be the indicator function of $A$. What will be the explicit description of $f^*$ in the Birkhoff ergodic theorem ?
Thank you for your help.
Davide, I believe this is how it works:
we observe that $\frac{1}{3k-1}\sum^{3k-1}_{i=0}f(T^i(x))=\frac{k}{3k-1}(f(x)+f(T(x)+f(T^2(x))$ , then from Birkhoff's Ergodic Throem taking the limit we have :
$f^*(x)=\frac{1}{3}(f(x)+f(T(x)+f(T^2(x))$, now we can use the definition of $f$ to write $f^*$ explicitly.