G Functions

We will investigate solutions G to a form of Schroeder's functional equation G(sz) = f(G(z)) where f is analytic at zero, f(0) = 0, and f '(0) = s where s has magnitude greater than one. (G is the inverse of the classical Schroeder function for f.) It follows that the set of functions which functionally commute with f are all those of the form h(z) = G(tG^-1(z)) where t is any complex number. In particular if t = sⁿ then h is the nth functional iterate of f. Setting t equal to a square root of s makes h a functional square root of f. (There are precisely two of them!)

For the functions plotted here, f was chosen to be a ratio of quadratic polynomials. These G functions have been approximated using the limit G(z) = lim fⁿ(z/sⁿ) where fⁿ denotes the nth iterate of f. All of the functions determined in this way are meromorphic. That is, in any bounded subset of the complex plane, G is analytic except for finitely many poles. As a measure of the complexity of these functions it is interesting to note that for many of those plotted here the number of poles in the plotted region may be a billion or more.

If f(a) = a, we call a a fixed point of f. If f(a) = b, f(b) = c and f(c) = a, the set {a,b,c} is called a fixed cycle of f of order three. If in addition, for any point z sufficiently close to a, f(f(f(z))) is even closer to a we say the cycle is an attracting cycle. If the derivative of f at a fixed point has magnitude less than one, f draws nearby points nearer and it is an attracting fixed point. If the derivative of f at the fixed point has magnitude greater than one nearby points are pushed away and we say the fixed point is repelling. Note that zero is a fixed point of f, and since the derivative of f at zero is greater than one in magnitude, zero is a repelling fixed point.

Functions with an attracting fixed point at zero but not a zero derivative also have a Schroeder function since the inverse of f has a repelling fixed point at zero. In that case the Schroeder function can be computed as the limit G^-1(z) = lim fⁿ(z)/sⁿ. Many years ago I plotted examples of the Schroeder function this way which gave me beautiful graphs although the Schroeder function in those examples was defined on a bounded set with a natural boundary which could not be extended. (If anyone pursues this I would love to see the results.) Some functions with a neutral fixed point (|s| = 1) have Schroeder functions and some do not. The current description of which category a function falls into is not completely satisfactory.

Prominent in many of these graphs are regions where the color is constant or almost constant. The colors of these regions correspond to the attracting fixed points and attracting fixed cycles of f(z). It should be pointed out that any meromorphic function which is constant on any line segment, no matter how short, is constant everywhere. Neither our eyes nor the computer display can reflect infinitesimal differences.

Since we are limiting f to be a quadratic rational function it can have a total of at most two attracting fixed points and cycles. (This is a rather deep theorem!)

The program which draws these graphs plots the value of fⁿ(z/sⁿ) where n is chosen large enough to make the magnitude of z/sⁿ smaller then a fixed epsilon > 0. Occasionally the graph may have noticeable discontinuities which can be eliminated by reducing the size of epsilon. The program also does a numerical search for attracting fixed points and cycles. As a result the points and cycles are only known approximately, and their existence has not been proven but assumed based on the numerical search and the behavior of G.