Project: Dynamics 2019

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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 $\mathbb R$ .

In [1]:

f(x) = x + sin(x)

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

In [2]:

plot(f,(x,0,20), aspect_ratio=1)

In [3]:

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:

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:

In [4]:

identity(x) = x

In [5]:

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

In [6]:

plt = cobweb(0.5, f, 10, 0, 3.5)
plt

In [7]:

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 $a<b$ 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:

In [8]:

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.

In [9]:

cobweb(3.0, f, 20, -5, 5)

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

In [10]:

forward_orbit(3.0, f, 20)

[3.00000000000000,
 -3.88999249660045,
 3.25721383591138,
 -4.15053714995653,
 3.71778289092915,
 -4.45632730660125,
 4.30305469756698,
 -4.60105339417880,
 4.58994767759334,
 -4.61208327222891,
 4.61194567936355,
 -4.61222017254902,
 4.61221879282579,
 -4.61222154535601,
 4.61222153155839,
 -4.61222155908446,
 4.61222155894648,
 -4.61222155922174,
 4.61222155922036,
 -4.61222155922312,
 4.61222155922310]

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.

In [11]:

g(x) = f(f(x))

In [12]:

plot(g, -5, 5, aspect_ratio=1) + plot(x, (x,-5, 5), color="red")

Homeomorphisms of R\mathbb RR

Orientation-preserving homeomorphisms

Orientation reversing homeomorphisms

Homeomorphisms of $\mathbb R$