Let $X$ be the space of all continuous maps from $S^{1}$ to $S^{1}$. $X$ is equipped with $C^{0}$ topology. Let $Y\subset X$ be the union of all finite order homeomorphisms i.e: all homeomorphisms $\phi$ with $\phi^{n}=Id$ for some natural number $n$. Assume that $g\in X$ belongs to $\bar{Y}-Y$. Some questions about such $g$:


1.Is $g$ necessarilly a homeomorphism?If yes, is it necessarilly order preserving?


2. Is it true to say that $g$ satisfies at least one of the following conditions:


I: The periodic points of $g$ is dense


II: $g$ has a dense orbit 


More generally, to what extent, the elements of $\bar{Y}-Y$ are known, dynamically?


What about if we replace $S^{1}$ with another manifold and study the elements of $\bar{Y}-Y$? What about if we consider $C^{k}$ topology for $k>0$?


**Note:** A possible operator theoretical version for question $1$ could be the following:Let $A$ be a $C^{\*}$ algebra. Let $X$ be the space of all unital morphisms and $Y\subset X$ be the union of all finite order automorphisms. Assume that $T\in \bar{Y}-Y$. Is $T$ necessarilly an automorphisms.(Actually, is it surjective?)



The closure of all periodic homeomorphisms of circle
The answer to both questions 1 and 2 is false, due to the following example.
Consider $(n+1)$ consecutive intervals $I\_0,...,I\_n$ of lengths $1,2,4,...,2^{n-1}, 2^n$. Let the map $T\_n$ cyclically permute them in an affine way. In fact, if you choose $I\_j=[2^j,2^{j+1}]$, then 
$$
T\_n(x)=\begin{cases}
2x, & x \not\in I\_{{n}}\\
2^{-n}x, & x\in I\_{n}.
\end{cases}
$$


Now, make an affine change of the coordinates so that the union $I\_0\cup\dots\cup I\_n=[1,2^{n+1}]$ becomes (always the same) circle $[0,1]/(0\sim 1)$. Then, the constructed maps on this circle $C^0$-converge, as $n\to\infty$, to
$$
T(x)=\begin{cases}
2x, & x \in [0,1/2]\\
0, & x\in [1/2,1].
\end{cases}
$$


This map is not a homeomorphism (it collapses the interval $[1/2,1]$ to a single point 0). Also, the orbit of any point falls to the fixed point 0 in a finite number of steps, so there is only one periodic point, and there is no dense orbit.


