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SharedPeriodicity.ipynbOpen in CoCalc

Periodicity

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

D(x) = 2*x - floor(2*x)
show(D)
x  2x2xx \ {\mapsto}\ 2 \, x - \left \lfloor 2 \, x \right \rfloor

Some example applications:

D(1/4)
1/2
D(2/3)
1/3

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

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:

plot(D,(x,0,1), aspect_ratio=1, exclude=[1/2,1])
# exclude corrects the plotting for discontinuities
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 DD is a value xx so that D(x)=xD(x)=x. Zero is a fixed point:

D(0)
0

A periodic point of DD is a point xx so that Dk(p)=pD^k(p)=p for some k1k \geq 1. The least period (or prime period) of pp is the smallest such kk. We say xx has period kk if Dk(x)=xD^k(x)=x.

The number 13\frac{1}{3} has least period two, since the following output shows that D(1/3)=2/3D(1/3)=2/3 and D2(1/3)=1/3D^2(1/3)=1/3.

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: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 (the cobweb path) [(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].

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

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:

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

The point 1/51/5 has period 44:

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

Here is another phenomenon. The point 1/61/6 is pre-periodic or eventually periodic. This means that there is a k>0k>0 so that Dk(1/6)D^k(1/6) is periodic. For 1/61/6, this kk is one since D(1/6)=1/3D(1/6)=1/3, and above we showed that 1/31/3 is period 22:

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

Lets compute the least period of 73/103:

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

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