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 $k>0$?

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?)

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

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 $k>0$?

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 $k>0$?