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 $\mathbb R$.
f(x) = x + sin(x)
Here we plot the function over the interval (0,10):
plot(f,(x,0,20), aspect_ratio=1)
def forward_orbit(x, T, N): ''' Return the list [x, T(x), ..., T^N(x)]. ''' orbit = [x] # Start of the orbit. y = x for i in range(N): y = T(y) # Redefine y to be T(y) orbit.append(y) # Add y at the end of the orbit. return orbit
A cobweb plot is a useful way to visualize an orbit of a map $T:{\mathbb R} \to {\mathbb R}$. It involves several things:
Here we define the identity map:
identity(x) = x
def cobweb(x, T, N, xmin, xmax): cobweb_path = [(x,x)] for i in range(N): y = T(x) # Reassign y to be T(x). cobweb_path.append( (x,y) ) cobweb_path.append( (y,y) ) x = y # Reassign x to be identical to y. cobweb_plot = line2d(cobweb_path, color="red", aspect_ratio=1) function_graph = plot(T, (xmin, xmax), color="blue") # define the identity map: identity(t) = t id_graph = plot(identity, (xmin, xmax), color="green") return cobweb_plot + function_graph + id_graph
plt = cobweb(0.5, f, 10, 0, 3.5) plt
plt = cobweb(6, f, 10, 0, 6.4) plt
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 $ax$. 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
A homeomorphism $T:\mathbb R \to \mathbb R$ is orientation-reversing if $x
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:
f(x) = cos(x) - x + 1/10 plot(f,-2*pi,2*pi, aspect_ratio=1)
The following is the cobweb plot of the orbit of 3.
cobweb(3.0, f, 20, -5, 5)
It seems to be approaching a period two orbit, which is further supported by the following:
forward_orbit(3.0, f, 20)
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.
g(x) = f(f(x))
plot(g, -5, 5, aspect_ratio=1) + plot(x, (x,-5, 5), color="red")