Other invariant measures than Lebesgue measure?

Question

Consider a rational rotation of the circle. What are other invariant measures different than the Lebesgue measure?

Any hints will be greatly appreciated!

You might be interested in log-polar coordinates. (A.k.a. how computer vision ignores rotation of the sought for item. Rotation in Cartesian space is translation in log-polar coordinates.) — Eric Towers, Sep 03 '24 at 16:15

Alp Uzman · Accepted Answer · 2024-09-03T22:21:17.173

4

Hint: 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 $X$.

edited Sep 03 '24 at 22:21

answered Sep 03 '24 at 20:50

Alp Uzman

is there is a reason why the average of Dirac measure has highest power $p-1$? – Emptymind Oct 17 '24 at 03:59

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