$T(z)=z^n$ preserves haar measure on the unit circle

Question

I'm studying Peter Walter's book Ergodic Theory and I faced an example which is difficult for me to prove:

$T(z) = z^n$ preserves Haar measure for $n\in \mathbb{Z}\setminus \{0\} $.

I searched on the net and found some useful information here: Theorem (4) Section 1.1. Walters' Ergodic Theory But I still don't know exactly what $T^{-1}(E)$ is. Why does it consist $n$ subintervals of length $E$? Could anyone make me understand this better?

Try to prove this for the open intervals first. If $E={z\colon\arg z\in(a,b)}$ for some $0\le a<b<2\pi$, then what is the preimage $T^{-1}(E)$? — richrow, Nov 14 '21 at 08:13
And do you know the computation of $n$th roots in the complex plane using polar coordinates? — GEdgar, Nov 15 '21 at 10:00

score 1 · Answer 1 · answered Nov 17 '21 at 07:40

1

Hint: The graph of a self-map of the circle (in additive notation) can be drawn on the torus, or alternatively on the unit square. To see the preimage of an interval, take an interval on the vertical axis and look at all the horizontal intervals mapped to it.

Here is a humble graph: https://www.desmos.com/calculator/nrnhipltl0

answered Nov 17 '21 at 07:40

Alp Uzman

10,742

More details here: https://math.stackexchange.com/q/4316072/169085 – Alp Uzman Dec 01 '21 at 03:50

$T(z)=z^n$ preserves haar measure on the unit circle

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