Periodicity

We consider the doubling map on the circle $\mathbb{R}/\mathbb{Z}$ :

In [1]:

D(x) = 2*x - floor(2*x)
show(D)

Some example applications:

In [2]:

D(1/4)

1/2

In [3]:

D(2/3)

1/3

Here we plot the function over the interval (0,1):

In [4]:

plot(D,(x,0,1), aspect_ratio=1)

As a remark, we can get rid of those vertical lines (which arise because Sage doesn't realize that the function is not continuous) using the exclude parameter:

In [5]:

plot(D,(x,0,1), aspect_ratio=1, exclude=[1/2,1])
# exclude corrects the plotting for discontinuities

In [6]:

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

Recall that a fixed point of $D$ is a value $x$ so that $D(x)=x$ . Zero is a fixed point:

In [7]:

D(0)

0

A periodic point of $D$ is a point $x$ so that $D^k(p)=p$ for some $k \geq 1$ . The least period (or prime period) of $p$ is the smallest such $k$ . We say $x$ has period $k$ if $D^k(x)=x$ .

The number $\frac{1}{3}$ has least period two, since the following output shows that $D(1/3)=2/3$ and $D^2(1/3)=1/3$ .

In [8]:

forward_orbit(1/3, D, 2)

[1/3, 2/3, 1/3]

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 (the cobweb path) $[(x,x), (x,T(x)), (T(x),T(x)), (T(x), T^2(x)), \ldots].$

The following function draws a cobweb plot of the orbit of $x$ , connecting $(x,x)$ to $\big(T^N(x),T^N(x)\big)$ by a cobweb path:

In [9]:

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

Here is the cobweb plot of 1/3:

In [10]:

plt = cobweb(1/3, D, 2, 0, 1)
plt

The point $1/5$ has period $4$ :

In [11]:

plt = cobweb(1/5, D, 4, 0, 1)
plt

Here is another phenomenon. The point $1/6$ is pre-periodic or eventually periodic. This means that there is a $k>0$ so that $D^k(1/6)$ is periodic. For $1/6$ , this $k$ is one since $D(1/6)=1/3$ , and above we showed that $1/3$ is period $2$ :

In [12]:

plt = cobweb(1/6, D, 3, 0, 1)
plt

Lets compute the least period of 73/103:

In [13]:

x = D(73/103)
k = 1
while x != 73/103:
    x = D(x)
    k = k+1
print( "73/103 has least period " + str(k) )

73/103 has least period 51

In [14]:

cobweb(73/103, D, 51, 0, 1)

Exercise: Think about why if $p/q$ is a fraction, then it must be either periodic or eventually periodic under $D$ . Under what conditions is it periodic? When is it eventually periodic?