Property of $f(x)=(3x+2\sin x) /_{2\pi \mathbb{Z}}$

Question

Consider $f:S^1\rightarrow S^1$ s.t. $$f(x)=(3x+2\sin x) /_{2\pi \mathbb{Z}}, \quad x \in S^1$$ How could one show that for any open (non-empty) set $A\subset S^1$ there exists $k\in \mathbb{N}$ s.t. $$f^{(k)}(A) \equiv f \circ f \dots \circ f (A)=S^1 \quad?$$

Answer 1

A pictorial representation of the map $f(x)$ is as below.

Let function $g(x)$ be defined with the same relation as $f(x)$, except the modular part. The point at which the first discontinuity happens is called $x_1$.

First, we consider the interval $[0,a)$, $a<x_1$. $0$ is a fix-point and ,because the derivative of $g(x)$ is larger than $1$, $g(a)>a$. So, the map $g(x)$ on the interval $[0,a)$ would widen the interval to $[0,g(a))$.

Now,

$f([0,a))=g([0,a))(mod 2\pi)$

If $g(a)>2\pi$, then the map covers $s^1$. Otherwise, $g(x)$ would be applied to the interval $[0,f(a))=[0,g(a))$ again. This process is done until the upper bound of the interval gets larger than $2\pi$.

Therefore, it is proved that $[0,a)$, $a<x_1$ would be the interval that we need, i.e, after a limited number of iterative mapping, it would cover $S^1$.

It is left to show that each interval $A=(a,b)$, such that$0<a<b<2\pi$, would contain $0$, after a limited number of iterative mapping.

First, we need to prove that the distance between $a$ and $b$ is increased, after every step of evaluating them with function $g(x)$.

$b-a=d$

$g(b)-g(a)=3b+2sin(b)-3a-2sin(a)=3(b-a)-2(sin(a)-sin(b))$

knowing the relation below

$sin(a)-sin(b)<b-a$

we get

$g(b)-g(a)>3d-2d=d$

So, $(a,b)$ is mapped to $(g(a),g(b))$ and it is continuous.

according to the picture, after applying $mod 2\pi$ operation to $(g(a),g(b))$, either it stays continuous, which means it does not contain a root of $f(x)$, and we should apply the mapping again or it would contain a root of $f(x)$. In that case, $(g(a),g(b))(mod 2\pi)=(f(a),0)\cup[0,f(b))$. having the set $[0,f(b))$, we can use the previous argument to show that it would, eventually, contains $S^1$, after a limited number of iterative mapping.