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I'm working out an exercise which make me question my comprehension of rotation number. The definition that was handled to me is the following:

Given a homeomorphism $f:S^1 \rightarrow S^1$ that preserves orientation and $F:R\rightarrow R$ a lift of $f$ the rotation number is defined as $\rho(f)=lim_{n\rightarrow_\infty} \frac{F^n(x)-x}{n} \bmod 1$

My question here is that I can write this down as $\rho(f)=lim_{n\rightarrow_\infty} \frac{F^n(x)-x}{n} \bmod 1 = lim_{n\rightarrow_\infty} \frac{F^n(x)}{n}-\frac{x}{n} \bmod 1 = lim_{n\rightarrow_\infty} \frac{F^n(x)}{n} \bmod 1$

which means I can get rid of the $x$ term used in the definition.

This means I can also understand $\rho(f) = lim_{n\rightarrow_\infty} \frac{F^n(x)}{n} \bmod 1$ as a definition of rotation number. But I was wondering if that was actually right, why would be defined substracting $\frac{x}{n}$ instead which looks "more complex".

Is that right? Am I missing something here?

Thanks.

J. W. Tanner
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user1880062
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1 Answers1

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The rotation number is an invariant of a homeomorphism $f\colon S^1 \to S^1$ which measures the "average angle from $x$ to $F(x)$ on the circle". (Since we are viewing $S^1$ as $\mathbb R/\mathbb Z$, for $s,t \in S^1$, the angle from $s$ to $t$ is just $t-s$.)

Now the most obvious way to define this would be as, say, $\int_{0}^1(F(t)-t)dt$, but one of the insights Poincare had was that this "average angle" should be detectable on each orbit of the iterates of $F$: if $x\in S^1$, then the angle between $x$ and $f(x)$ is just $F(x)-x$, the angle between $f(x)$ and $f(f(x))$ is then $F(F(x))-F(x)=F^2(x)-F(x)$ and the angle between the third and second iterates of $f$ on $x$ is $F^{3}(x)-F^{2}(x)$, and so on. The reason for the lift to $F\colon \mathbb R\to \mathbb R$ is that it ensures we measure the angles between each iterate and the next in the same orientation (e.g. if $f(x) = x+1/2$ then this is measured as $+1/2$ rather than $-1/2$).

Thus we see that the average angle between the a point and its image under $f$ for the first $n$ points in the orbit of $x$ is $$ \begin{split} \frac{1}{n}\sum_{k=0}^{n-1}\left( F^{k+1}(x)-F^{k}(x) \right) &=\frac{1}{n} \sum_{k=0}^{n-1} F^{k+1}(x) - \sum_{k=0}^{n-1} F^k(x)\\ &= \frac{F^{n}(x)-x}{n} \end{split} $$

Thus although the definition $\lim_{n \to \infty}\frac{F^n(x)-x}{n} = \lim_{n \to \infty} \frac{F^n(x)}{n}$ the latter expression is not the quantity whose limiting behaviour was of interest to us.

krm2233
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