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Homeomorphisms of R\mathbb R

Orientation-preserving homeomorphisms

A homeomorphism of a topological space XX is a continuous map h:XXh:X \to X which has a continuous inverse h1:XXh^{-1}:X \to X.

In advanced calculus, you should have learned that a continuous map h:RRh:\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:RRh:\mathbb R \to \mathbb R is orientation-preserving if x<yx < y implies h(x)<h(y)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 R\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:RRT:{\mathbb R} \to {\mathbb R}. It involves several things:

  • The graph of the function ff.
  • 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),T2(x)),].[(x,x), (x,T(x)), (T(x),T(x)), (T(x), T^2(x)), \ldots].

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

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

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

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

Theorem.

  • Let aR{}a \in {\mathbb R} \cup \{-\infty\} and bR{+}b \in {\mathbb R} \cup \{+\infty\} be fixed points with $ax.Then. Then (a,b) \subset W^s(b)$.

  • Let aR{}a \in {\mathbb R} \cup \{-\infty\} and bR{+}b \in {\mathbb R} \cup \{+\infty\} be fixed points with $a

Orientation reversing homeomorphisms

A homeomorphism T:RRT:\mathbb R \to \mathbb R is orientation-reversing if $xT(y)$.

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

We have the following result:

Proposition. If T:RRT:\mathbb R \to \mathbb R is an orientation-reversing homeomorphism, then TT has a unique fixed point in R\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)
[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:RRT:\mathbb R \to \mathbb R is an orientation-reversing homeomorphism, then T2:RRT^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")