Homeomorphisms of $\mathbb R$

Orientation-preserving homeomorphisms

A homeomorphism of a topological space $X$ is a continuous map $h:X \to X$ which has a continuous inverse $h^{-1}:X \to X$.

In advanced calculus, you should have learned that a continuous map $h:\mathbb R \to \mathbb R$ is a homeomorphism if and only if it is one-to-one and onto. (This does not hold for all spaces!)

A homeomorphism $h:\mathbb R \to \mathbb R$ is orientation-preserving if $x < y$ implies $h(x) < h(y)$.

Since the following function has derivative which is non-negative and not identically zero on any interval, it is a homeomorphism of $\mathbb R$.

Here we plot the function over the interval (0,10):

A cobweb plot is a useful way to visualize an orbit of a map $T:{\mathbb R} \to {\mathbb R}$. It involves several things:

• The graph of the function $f$.

• The diagonal (the graph of the identity map)

• The orbit. The orbit is visualized as the sequence of points $$[(x,x), (x,T(x)), (T(x),T(x)), (T(x), T^2(x)), \ldots].$$

Here we define the identity map:

The stable set of a point periodic point $p$ is the set of points $x$ so that $$\lim_{n \to \infty} dist\big(T^n(p), T^n(x)\big)=0.$$ The set $W^s(p)$ denotes the stable set of $p$. We can see from the above cobweb plots that for $T(x)=x+sin(x)$, we have that $W^s(\pi)$ is the open interval $(0,2 \pi)$.

If $x \in W^s(p)$ we say $x$ is forward asymptotic to $p$.

We remark that if $T:\mathbb R \to \mathbb R$ is an orientation-preserving homeomorphism, we can continuously extend the definition of $T$ so that $T(+\infty)=+\infty$ and $T(-\infty)=-\infty$.

The following theorem completely describes the longterm behavior of orienation preserving homeomorphisms:

Theorem.

• Let $a \in {\mathbb R} \cup \{-\infty\}$ and $b \in {\mathbb R} \cup \{+\infty\}$ be fixed points with $a<b$ such that for every $x \in (a,b)$, $T(x)>x$. Then $(a,b) \subset W^s(b)$.

• Let $a \in {\mathbb R} \cup \{-\infty\}$ and $b \in {\mathbb R} \cup \{+\infty\}$ be fixed points with $a<b$ such that for every $x \in (a,b)$, $T(x)<x$. Then $(a,b) \subset W^s(a)$.

Orientation reversing homeomorphisms

A homeomorphism $T:\mathbb R \to \mathbb R$ is orientation-reversing if $x<y$ implies $T(x)>T(y)$.

If $T:\mathbb R \to \mathbb R$ is an orientation reversing homeomorphism then $T$ extends so that $T(+\infty)=-\infty$ and $T(-\infty)=+\infty$.

We have the following result:

Proposition. If $T:\mathbb R \to \mathbb R$ is an orientation-reversing homeomorphism, then $T$ has a unique fixed point in $\mathbb R$.

Existence of a fixed point is a consequence of the Intermediate Value Theorem. Uniqueness is a consequence of the definition of orientation-reversing.

Consider the following example:

The following is the cobweb plot of the orbit of 3.

It seems to be approaching a period two orbit, which is further supported by the following:

Also observe that:

Proposition. If $T:\mathbb R \to \mathbb R$ is an orientation-reversing homeomorphism, then $T^2:\mathbb R \to \mathbb R$ is an orientation-preserving homeomorphism.