Something dynamics can tell us about number theory

Review of cycles

The fixed points of f can be found by a simple geometric property: they are the intersections of the graph of f and the diagonal.

Cycles of f can be detected geometrically, by observing that points of an n-cycle of f are fixed points of fn.

For example, suppose {x1, x2} is a 2-cycle for f. That is, f(x1) = x2 and f(x2) = x1. Then

f2(x1) = f(f(x1)) = f(x2) = x1

Suppose x* is a fixed point of f2. Then x* might

belong to a 2-cycle of f, or

be a fixed point of f: if f(x*) = x*, then f2(x*) = x*.

Suppose x* is a fixed point of f3. Then x* might

belong to a 3-cycle of f, or

be a fixed point of f

Suppose x* is a fixed point of f4. Then x* might

belong to a 4-cycle of f, or

belong to a 2-cycle of f: if f2(x*) = x*, then f4(x*) = f2(f2(x*)) = f2(x*) = x*

be a fixed point of f

In general, if x* is a fixed point of fn, then x* belongs to an m-cycle for some m that divides n.

As a consequence, if x* is a fixed point of fp and p is prime, then x* belongs to a p-cycle of f, or is a fixed point of f.

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