Does this limit always exist? Here is the question; it may seem very simple, but it is difficult (at least for me).
Let $f(x)$ be a continuous function on $R$ that is strictly increasing, and suppose $g(x)=f(x)-x$ is a periodic function with period 1.
Prove that for all $x\in R$, $\lim_{n\to \infty}\frac{f^n(x)}{n}$ exists.
In an equivalent formulation, the dynamic system $(X,T)$ is quasi-regular, where $X=[0,1],T: x\mapsto x+\{g(x)\}$. I.e. the Birkhoff average $\frac{1}{N}\sum_{n\leq N}T^n(x)$ exists for all $x\in X$.
 A: EDITED: below I give a counter example in the case where $f$ is not monotonic. The question you’re asking is well known to be true in the monotonic case
There's a counterexample (even if $f$ is highly regular). I'll give a piecewise linear counterexample: let 
$$
f(x)=\begin{cases}4x&\text{if $x\in [0,\frac 12)$;}\\
3-2x&\text{if $x\in [\frac 12,1)$};
\end{cases}
$$
and extend to the real line by periodicity. 
Let $g(x)$ be the corresponding map from the circle to itself. 
Now let $J_0=[0,\frac 14)$ and $J_1=[\frac 14,\frac 12)$. Both of these intervals map bijectively to the whole circle. Now you can define 
$J_{i_0\ldots i_{n-1}}=J_{i_0}\cap g^{-1}J_{i_1}\cap \ldots\cap g^{-(n-1)}J_{i_{n-1}}$. 
Finally for any sequence of 0's and 1's, you can define $h(z)=\bigcap_{n=0}^\infty J_{z_0\ldots z_n}$. For $x\in\mathbb R$, let $\rho^+(x)=\limsup f^n(x)/n$ and $\rho^-(x)=\liminf f^n(x)/n$. It's not hard to check that $\rho^+(h(z))=\limsup (z_0+\ldots+z_{n-1})/n$ and $\rho^-(h(z))=\liminf (z_0+\ldots+z_{n-1})/n$. In particular, it's easy to construct points with $\rho^-(x)=a$ and $\rho^+(x)=b$ for any $0\le a\le b\le 1$.
