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Let $T: S^1 \to S^1$ be the rotation of the circle by $90$ degrees. I want to find an ergodic T-invariant measure.

My question is:

I was looking here Other invariant measures than Lebesgue measure? and I thought that the measure given in this link might work.

Here is the measure given in the link above:

If $f:X\to X$ is a function and $x\in X$ is such that $f^p(x)=x$ for some $p\in\mathbb{Z}_{\geq1}$, then the average of Dirac measures

$$ \dfrac{\delta_x+\delta_{f(x)}+\cdots+\delta_{f^{p-1}(x)}}{p} $$

is an $f$-invariant Borel probability measure on $X$.

Am I correct? If so, I am not sure why this measure is ergodic though and why it is $T$- invariant? And why it can not be lebesgue?

Can someone explain this to me please?

If not, could you please direct me to the right direction of thinking?

Emptymind
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1 Answers1

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Am I correct?

Indeed. The way you are parameterizing the circle, $90$ degree rotation is a rational rotation ($90^o=2\pi q$ for some rational number $q$).

If so, I am not sure why this measure is ergodic though and why it is $T$-invariant?

Call $\mu_x$ the measure in question. Then for $\phi:X\to\mathbb{R}$ a function one has

\begin{align*} \int \phi\circ f(y) d\mu_x(y) = \dfrac{\phi\circ f(x)+\phi\circ f(f(x))+\cdots+\phi\circ f(f^{p-2}(x))+\phi\circ f(f^{p-1}(x))}{p} = \dfrac{\phi\circ f(x)+\phi\circ f^2(x)+\cdots+\phi\circ f^{p-1}(x)+\phi\circ f^{p}(x)}{p} = \dfrac{\phi\circ f(x)+\phi\circ f^2(x)+\cdots+\phi\circ f^{p-1}(x)+\phi(x)}{p} =\int\phi d\mu_x(y), \end{align*}

so that $\mu_x$ is $f$-invariant. Finally for ergodicity, let $A$ be an $f$-invariant subset. Either $x\in A$ xor $x\not\in A$. In the first case, by $f$-invariance of $A$, the whole trajectory of $x$ must be in $A$, hence $\mu_x(A)=1$. In the second case, by a similar reasoning the whole trajectory of $x$ must miss $A$, hence $\mu_x(A)=0$.

And why it can not be lebesgue?

Lebesgue measure is fully supported, that is, the lebesgue measure of any nonempty open subset is positive. On the other hand, there are open sets (e.g. open arcs) that the trajectory of $x$ misses.

Alp Uzman
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