Rotations of a circle

To give further illustrations of the possible behavior of sequences, we will next consider the dynamical system of rotating a circle through a fixed angle. We will use Euler's formula to describe points on the unit circle in the complex plane (see this section. The angle is determined by the point only up to the addition of an integer multiple of . Our metric space is

with the usual distance formula in the plane. Fix an angle and define

(Rotate by counterclockwise.) The iterates are

A picture of these iterates is shown below.

  
Figure 36: Rotating an angle

The behavior of this sequence depends on the choice of . If is a rational multiple of , say , p, q integers, then . Then , since . In other words, rotation by performed 2q times in a row fixes all points on the circle. Thus, the sequence of iterates repeats periodically

The attractor of the orbit is the finite set of points

which does depend on the initial point . However, for a different axiom , the whole orbit is simply rotated through the angle .

What happens if is irrational? Then the sequence never repeats; the question is how does it wander around the unit circle. This is easy to explore by means of a few computer experiments. First, we simplify the dynamical system a little. Instead of the complex numbers , we keep track only of the angle . We require angles to belong to the interval , and we reduce by multiples of when necessary to achieve this. For convenience, we take as the starting angle. Finally, we replace by , and compute the iterates modulo multiples of 1, instead of . Thus, the n-th iterate is

where denotes the greatest integer less than or equal to y, commonly called the ``floor'' of y. is called the ``fractional part'' of y.

This is a very easy procedure to encode in MAPLE, and we have done just that in the worksheet:

circle.ms

Here is the listing of the procedure angles that generates the list of iterates . The key routine is MAPLE's builtin procedure frac for computing fractional parts.

  angles:= proc(x,n)
      local y,v,i; 
      y:= frac(x);
      v:=[y];
      for i from 1 to n do
            y:=  frac(y+x);
            v:=  [op(v), y]
      od
  end:

The command angles(.6,10) displays the first ten iterates with x=0.6

We see that the sequence repeats every five terms, which we can conclude from the fact that .6=3/5. How about , which the Greeks discovered (to their horror) is irrational? A small table of the iterates is shown below.

 

n n n n

1 .414 2 .828 3 .243 4 .657

5 .071 6 .485 7 .899 8 .314

9 .728 10 .142 11 .556 12 .970

13 .385 14 .799 15 .213 16 .627

17 .041 18 .456 19 .870 20 .284

21 .698 22 .112 23 .527 24 .941

25 .355 26 .769 27 .183 28 .598

 

Apparently, the iterates wander all over the unit interval. The study of the precise behavior of these orbits belongs to a fascinating field of mathematics called diophantine approximation. This subject is concerned with the approximation of arbitrary real numbers by rational numbers. The first theorem proved in this subject is

Kronecker's Theorem: (See [HW79], ch. XXIII.) If x is irrational, then every point in [0,1) is a limit point of the sequence .

This theorem does not say how to find a subsequence that approaches any given limit point, and in general this can be difficult to calculate. For , by examining our table and calculating a little further we might choose

as the beginning of a subsequence approaching .5. How to systematically choose this sequence is a difficult problem.

For irrational the attractor of rotation by is the whole circle. frequently, an attractor of a dynamical system has a ``fractal'' nature. We believe the term fractal should be reserved for ``irregular sets'' that arise as the attractors of a dynamical system. Understanding the dynamics is always a central goal of any field of natural science. However, out of the need to measure things, various methods have been invented for assigning a ``fractal dimension'' to any subset of a metric space. To the description of this dimension we devote the remainder of this chapter.