A rational function (of a complex variable z) is a ratio of polynomials

φ(z) = F(z)/G(z) = [a_d z^d + a_d-1 z^d-1 + … + a₁ z + a₀]/[b_e z^e + b_e-1 z^e-1 + … + b₁ z + b₀] .

For the moment we consider only rational functions with coefficients in the rationals Q. We assume, of course, that a_d and b_e are not zero. The degree of φ is the greater of d and e. We generally assume that the degree of φ is at least 2, just because the degree 1 case is straightforward to understand. In the subject of dynamical systems, we consider iterations of the function φ applied to a point z:

φⁿ(z) ≡ φ ο φ ο φ ο … ο φ(z) ,

where there are n iterations of the function φ. If α is a given complex number then we may look at its orbit

O_φ(α) = { α, φ(α), φ²(α), φ³(α), … } .

One studies the iterates of φ in part by classifying and describing the different sorts of orbits. One says that a point p is fixed by φ if φ(p) = p.

It is always helpful, in apprehending a new idea, to consider a concrete example. Many of the key ideas here are already interesting in the case that φ is a polynomial (i.e., G is a polynomial of degree 0).

EXAMPLE 1: Let

φ(z) ≡ z² = z²/1 .

Then we may make the following observations:

• Some points have orbits that "head out to infinity." For example,

  O_φ(2) = {2, 4, 16, 256, … } .

• Other points have orbits that "head in to zero." For example,

  O_φ(1/2) = { 1/2 , 1/4 , 1/16 , … } .

• Some points are fixed: for instance, φ(0) = 0 and φ(1) = 1. We may see these assertions more dynamically by writing

  O_φ(0) = {0, 0, 0, …} and O_φ(1) = {1, 1, 1, … } .

• Other points are "eventually fixed" in the sense that the original domain point is not fixed but instead some subsequent image point is fixed. As an instance,

  O_φ(-1) = {-1, 1, 1, 1, 1, …} .

• If we pass to complex domain elements, then we find points that "cycle". An instance is

  O_φ ( [-1 + √3 i]/2 ) = { [- 1 + √3 i]/2 , [- 1 - √3 i]/2 , [- 1 + √3 i]/2 , [- 1 - √3 i]/2 , … } .

Number theorists and dynamicists are particularly interested in studying periodic and preperiodic points of iterations of a rational function. A point α is called periodic if there is an n ≥ 1 such that φⁿ(α) = α. The least such n is termed the period of the point α. If the period is 1 then the point α is a fixed point. A point α is preperiodic if some iterate φⁱ(α) is periodic. [Equivalently, α is preperiodic if its orbit O_φ(α) is finite.] By contrast, a point β that has infinite orbit is called a wandering point.

EXAMPLE 2: Let us return to the function φ(z) = z² from Example 1. We see that

• The points 2 and 1/2 are wandering points.
• The points 0 and 1 are fixed points.
• The point [-1 + √3 i]/2 is a periodic point of period 2.
• The point -1 is a preperiodic point that is not a periodic point.

Number theorists would like to count the periodic points. It is not difficult to show that there are always infinitely many complex periodic points. In many interesting examples there are infinitely many real periodic points. But we would like to know how frequently rational periodic points can occur. It is a theorem of Northcott from the 1950s that there can only be finitely many rational preperiodic points. The main question considered at this AIM workshop is:

Can we bound the number of preperiodic rational points in terms of the degree of the rational function φ?

It is not even known whether there is such a bound, but the consensus (i.e., the conjecture) is that there is one. People are also interested in more general versions of this question, in which rational functions φ of several variables are allowed, and also we replace the rational field Q by a number field. In that situation, the desired upper bound on the number of preperiodic points would be in terms of (i) the degree of the rational function, (ii) the number of variables, and (iii) the degree of the number field.

The answer to the main question is not even known in degree 2 (in fact not even for polynomials of degree 2!), and that special case is the focus of this workshop. These results are analogous to a celebrated theorem of Barry Mazur that gives a uniform bound on the number of rational torsion points on an elliptic curve. Thus we see (at least philosophically) the connection with number theory. It turns out that the conjecture for rational functions of degree 4 implies (in just a few lines) Mazur's deep wtheorem.

About one third of the workers worldwide in this relatively new field of mathematical inquiry were present at the workshop. Techniques were shared, and new points of view offered. For example, in a lecture on Wednesday, Robert Benedetto indicated that one could bound the number of rational preperiodic points by the number of primes with bad reduction. The connection between number theory (a very old subject, going back to the ancient Greeks) and dynamical systems (a fairly new area, going back only to Henri Poincaré) is an exciting one that is leading to many new ideas.