Periodicity

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

Some example applications:

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

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:

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

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 $D^2(1/3)=1/3$.

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:

Here is the cobweb plot of 1/3:

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

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 $2$:

Lets compute the least period of 73/103:

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?