Activity 2-2: Mapping a set to itself

Let’s suppose we are given a mapping f, so that the list maps to f(x 1 ), f(x 2 ), f(x 3 ),... Consider an infinite set S of points that can be put into an infinite list, x 1, x 2, x 3... Now suppose f is such that one point goes to itself, two points are in a cycle (or orbit) of length 2, three are in a cycle of length 3 and so on. So if f(x) = x 2, we get the list x 1 2, x 2 2, x 3 2,...

These cycles generate an infinite sequence as follows: There is one point with period 1. There are four with period 2, but we need to include the point of period 1 here, as that returns to itself after being transformed twice, so that makes five points. What about period three? Nine points plus the period 1 point again; 10 points. Period 4 gives 16 points, plus the four period 2 points, plus the period 1 point, so that’s 21.

So this situation has created the infinite sequence starting 1, 5, 10, 21, 26, 50,... (the rule generating this might be quite complicated!) We can say that this sequence is realisable; we can turn it into a picture of points with their orbits. If you create a sequence that is realisable, It is a good idea to check it on the Online Encyclopedia of Integer Sequences. OEIS Address

So 1 = = = = What happens if we search for 1, 5, 10, 21, 26, 50,... ?

Task: try some sequences out and see if they correspond to orbit pictures like the ones above. Which of the sequences that we know from our everyday mathematical life are realisable? Task: try some orbit pictures out and see what sequences they generate. OR...

Fibonacci? The sequence of squares? The Lucas sequence has the same rule as the Fibonacci sequence, but starts 1, 3... Worth a try... Edouard Lucas, French, ( )

The Lucas sequence IS realisable; 1, 3, 4, 7, 11,....

Let’s take the natural numbers as our list, 0, 1, 2, 3... The function f takes the natural numbers to themselves as follows; Every odd number goes to its double. Every number divisible by 4 goes to itself. Every other even number goes to half itself. Here is another f that we might consider.

What happens to the natural numbers as f is repeated? Which points go to themselves (have period 1)? Which points go around in a cycle (have a period bigger than 1)? Which periods are possible here? Draw an orbit diagram for this f.

We have here a bijection; each number goes to a unique number, and each number is ‘hit’ by a unique number. We say the mapping is 1-1 (one-to-one.) This means that the inverse function exists – we can go backwards if we want to. Try to think of a different bijection, where a different range of periods are possible. Task: what is the inverse function here? If a bijection exists between two sets, then they are of the same size.

With thanks to: Tom Ward Carom is written by Jonny Griffiths,