Dynamical system with only one chain component Let $(X,d)$ be a compact metric space and $f:X\to X$ be a surjective map. 
A finite sequence $\{x_n\}_{n=0}^{k}$ is called $\varepsilon$-chain if $d(f(x_n), x_{n+1})<\varepsilon$ for $n=0, \ldots k-1$.
We say that the point $x$ is chain recurrent if for every $\epsilon>0$, there is $\varepsilon$-chain $\{x_n\}_{n=0}^{k}$ with $x_0=x_k=x$; and denote by $R(f)$ el set of chain recurrent points.
I am interested in the veracity of the following statement: If $R(f)$ is connected, then $X=R(f)$. 
I would greatly appreciate some counterexample or suggestion for the proof of it.
 A: I think it is true and can be proved along the following lines:


*

*For $n \in \mathbb{Z}_+$ put $p_n = f^{(n)}(x)$. By compactness, the
sequence $\{p_n\}$ has a cluster point $p$. From $\{p_n\}$ we can construct
$\delta$-chains showing that $p$ is chain-recurrent, i.e. $p \in R(f)$.
Also, for any given $\varepsilon$, a sufficiently long initial segment of
$\{p_n\}$ gives us an $\varepsilon$-chain from $x$ to $p$.

*Since $f$ is surjective, there is a sequence $\{q_n\}_{n=0}^\infty$
such that $f(q_{n+1}) = q_n$ and $q_0 = x$. Again, $\{q_n\}$ has a cluster point $q$ and 
again, $q\in R(f)$. This time reversing an initial segment of $\{q_n\}$
gives us an $\varepsilon$-chain from $q$ to $x$.

*Since $R(f)$ is connected, there is a family of points 
$\{h_i\}_{i=0}^m$ in $R(f)$ such that $h_0 = p$, $h_m = q$ and $d(h_i, h_{i+1}) < \frac{\varepsilon}{2}$ for all $i < m$. Taking a $\frac{\varepsilon}{2}$-chain from $h_i$ to itself and changing the
last point to $h_{i+1}$ results in an $\varepsilon$-chain from $h_i$ to 
$h_{i+1}$. Concatenating these chains yields an $\varepsilon$-chain
from $p$ to $q$.
Combining the chains from 1, 3 and 2, we have an $\varepsilon$-chain
from $x$ to itself for arbitrary $x\in X$ and $\varepsilon > 0$.
Therefore $R(f) = X$.
This may seem a very roundabout way of doing things. To see why this
is necessary, consider a case where $X$ is a circle and $f$ is a
homeomorphism where the fixed points form a semicircle. An 
$\varepsilon$-chain from a point to itself may really have to go
all the way around the circle, taking many "hops" between fixed points.
This is also an example where all points are chain-recurrent, but only
the fixed points are recurrent.
