As a motivattion for an introdutory notion in our Ergodic lecuture, we were asked to give a Geometric meaning to the following limit. Let $\lambda\in \Bbb T$

$$\lim_{n\to \infty}\frac1n\sum_{k=0}^{n-1} \lambda ^k= \begin{cases}1&\text{if}~\lambda=1 \\0 &\text{if}~\lambda \neq 1,\end{cases}$$

Where $\Bbb T= \{ e^{ia}: a\in \Bbb R\}$. Note that, difficult to see that

$$ \left| \frac1n\sum_{k=0}^{n-1} \lambda ^k \right|= \left|\frac1n\frac{\lambda^n-1}{\lambda-1}\right|\leq \frac{2}{n|\lambda-1|} \to 0$$

Does any body have clue on how to answer this?

  • $\begingroup$ Why are you calling it a torus? It is the unit circle. If $\lambda\ne1$ then $\lambda^n$ either traces out a regular polygon of points or it is dense in the unit circle. In either case, the average sum is $0$. $\endgroup$ – Chrystomath Mar 20 at 12:04

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