fixed points and their stability

Another characteristic of polynomials is their fixed points. They are points for which

P(z) = z (98)

By defining an auxiliary polynomial Q(z)=P(z)-z, the problem of finding fixed points is turned into finding zeroes, to which the previous analysis applies. Although zeroes and fixed points are expected to be different, their polynomials both have the same symmetric functions, with the exception of c_n-1, which is the coefficient of z and which differs by 1 between the two polynomials.

Zeroes play an important role in the multiplication of polynomials whereas fixed points dominate the iteration of polynomials. Besides identifying points where nothing changes on account of a polynomial mapping, the derivative at the fixed point describes the behavior of nearby points. That is because a Taylor's series expansion about the fixed point calls for a contraction or an expansion according to the absolute value of the derivative, extending over a region whose size depends on the remaining terms in the series.

According to the size of the derivative, fixed points are classified as

$$\begin{array}{ll} {\rm unstable} & \vert f'\vert > 1 \\ {... ...rt < 1 \\ {\rm superstable} & \vert f'\vert = 0 \end{array}$$

The principal characteristics of iteration can be summarized as:

1.

Fixed points persist,

2.

Critical points proliferate,

3.

Stability becomes ever more pronounced,

4.

Critical points respond to fixed point stability.

For the first item it suffices to note that if P(x) = z, then P(P(z)) = P(z) = z and so on for any further application of P.

For the second item, the chain rule for derivatives is pertinent. Set G(z)=P(P(z)) and observe that

 

$$\frac{d}{dz}P(P(z)) P'(P(z))\ P'(z).$$ = (99)

Therefore, if z is a point for which P'(z)=0, its effect as the right hand factor in equation (99) is to make G'(z)=0, indicating a critical point for the composite function. So it is that critical points persist, just as fixed points do.

But suppose that z maps into a point P(z)=y for which P'(y)=0. Then G'(z)=0 because of the left hand factor in equation (99). So all the counterimages of any of the function's critical points turn up as critical points for the iterated function, which accounts for the claimed proliferation.

 

Figure 9: critical points and fixed points of a map and its composite.

The chain rule also describes the behavior of the derivatives considered as multipliers at fixed points. Namely, their values at each point in a trajectory $z, P(z), P(P(z)), \ldots$ multiply. If the point is fixed this product is a power and so a stable fixed point becomes ever more stable, neutral points remain neutral, and unstable points become increasingly unstable. This accounts for the intensification of the stability parameters of as fixed point, but just looking at a formula does not always convey the severity of its consequences. A factor as small as 2 becomes 1,024 after just ten iterations.

The extreme flattening of a function around its stable fixed points and accompanying steepness in the vicinity of unstable fixed points reflects itself in attracting critical points towards the stable points and repelling them from unstable points.

Imaging and counterimaging define trajectories in the complex plane. There is no reason for a series of images to close, but if it does, then there is a sequence of values which must repeat forever after.

The part which repeats is a cycle, the sequence leading into the cycle is a transient. The cycle is stable or not according to the magnitude of the product of all the derivatives taken around the cycle, the same whatever the reference point as long as the complete cycle is consulted.

There is likewise no guarantee that the set of counterimages, even of a fixed point or of a cycle, closes. Generally it will not, especially in view of the number of roots of the polynomial establishing the counterimage. In fact, the collection of counterimages will resemble a rooted tree, which offers the opportunity of introducing an invariant polar coordinate system for the tree. The radius is the number of iterations, the angle divides the previous arc by the number of roots and puts them down in some order - say by their phases.

The counterimage tree, carried to extreme, is infinite. However it doesn't fill up the plane, but more nearly an image of the unit circle.

For an unstable fixed point this collection of counterimages is called a Julia set. The classical studies of iteration were performed around the year 1920 by Gaston Julia and the somewhat older Pierre Fatou, and were related to a prize offered by the French Academy for an earlier problem of Cayley. In the forties, Carl Ludwig Siegel made an important contribution to the theory of neutral fixed points, but still more recently computer processing and computer graphics have permitted experimental work which has stimulated the theory even further, beginning with the work of Benoit Mandelbrot during the seventies.