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It is a rough survey with few precise statements or proofs, and depends strongly on work by Douady, Hubbard, Branner and Rees.
The parameter space for cubic maps.
Following Branner and Hubbard, any cubic polynomial map from the complex numbers C to C is conjugate, under a complex affine change of variable, to a map of the form with critical points EQU.
(Compare Appendix A.)
This normal form is unique up to the involution which carries to the map EQU, changing the sign of EQU.
Thus the two numbers form a complete set of coordinates for the moduli space, consisting of complex cubic maps up to affine conjugation.
The invariant EQU can be thought of as a kind of discriminant, which vanishes if and only if the two critical points coincide; while EQU is a measure of asymmetry, which vanishes if and only if EQU is an odd function.
Now consider a cubic map EQU with real coefficients.
If we reduce to normal form by a complex change of coordinates, as above, then we obtain a complete set of invariants EQU which turn out to be real.
However, if we allow only a real change of coordinates, then there is one additional invariant, namely the sign of the leading coefficient.
It is not difficult to check that EQU coincides with the sign EQU whenever EQU.
However, this additional invariant EQU is essential when EQU, for in this case there are two essentially different real polynomial maps which are conjugate over the complex numbers, but not over the real numbers.
Thus the moduli space of real affine conjugacy classes of real cubic maps can be described as the disjoint union of two closed half-planes, namely the half- plane EQU and the half-plane EQU.
Any real cubic map is real affinely conjugate to one and only one map in the normal form (When EQU, we can use the alternate normal form EQU.)
In the two quadrants where EQU, note that the associated real cubic map has real critical points, while in the remaining two quadrants, EQU, it has complex conjugate critical points.
Further details may be found in Appendix A.
Real cubic maps as real dynamical systems.
Let us try to describe the behavior of the iterates of a cubic map EQU, considered as a real dynamical system.
It is convenient to introduce the notation EQU for the compact set consisting of all points EQU for which the orbit EQU is bounded.
This set EQU can be described as the real part of the “filled Julia set” of EQU.
Figure 1.
Representative graphs for the four different classes of real cubic polynomials.
The case EQU is illustrated; but corresponding examples with EQU can be obtained by looking at this figure in a mirror.
We first introduce a very rough partition of each parameter half-plane for real cubics into four regions EQU, EQU, EQU and EQU.
More generally, we divide real polynomial maps EQU of degree EQU into EQU distinct classes EQU, as follows.
We will say that EQU belongs to the trivial class EQU if EQU consists of at most a single point.
(More precisely, EQU will consist of one fixed point when the degree is odd, and will be vacuous when the degree is even.)
If EQU does not belong to this trivial class, then there must be at least two distinct points in EQU.
Let EQU be the smallest closed interval which contains EQU.
Thus every orbit which starts outside of EQU must escape to infinity, but the two end points of EQU must have bounded orbits.
In fact, it follows by continuity that each endpoint of EQU must map to an endpoint of EQU.
For EQU, we will say that EQU belongs to the class EQU if the graph of EQU intersected with EQU has EQU distinct components.
In other words, EQU belongs to EQU if the interval EQU can be partitioned into EQU closed subintervals which map into EQU (some of these intervals may be degenerate when EQU ), together with EQU separating open intervals which map strictly outside of EQU.
Note that EQU, since each of these open intervals must contain a critical point of EQU.
As an example, for degree EQU the quadratic map EQU belongs to For any degree EQU, note that EQU belongs to the class EQU if and only if the compact set EQU is a non-trivial interval (coinciding with EQU ), or in other words if and only if this interval EQU maps into itself, with all orbits outside of EQU escaping to infinity.
For EQU in EQU with EQU at least EQU of the critical orbits, that is the orbits of the critical points, must be real and must escape to infinity.
The case EQU is of particular interest.
If EQU, then all of the critical orbits escape to infinity.
Furthermore, the interval EQU contains EQU disjoint subintervals, each of which is mapped diffeomorphically onto the entire interval EQU.
A rather delicate argument, following, then shows that the set EQU is a Cantor set of measure zero.
Furthermore, the restriction EQU is homeomorphic to a one-sided shift on EQU symbols.
The degree EQU polynomials in EQU have maximal topological entropy, equal to EQU.
They have the property that their complex periodic points are all distinct and contained in the real interval EQU.It follows that their (complex) Julia set coincides with the Cantor set EQU.
We now specialize to the cubic case EQU.
In order to separate the four classes of real cubic maps, we introduce four curves in the parameter plane, as follows (Figure 2).
fig2.ps pixels 1152 by 900 scaled 225 Figure 2.
The four regions in the EQU parameter plane, and the curves which separate them.
Let EQU consist of all parameter pairs EQU for which the associated cubic map EQU has a periodic orbit of period EQU with multiplier EQU equal to EQU.
In particular, the curve consists of all parameter pairs for which the graph of EQU is tangent to the diagonal, while gives maps for which the graph of EQU is tangent to the diagonal.
Such points of tangency are called saddle nodes of period 1 or 2 respectively.
Similarly, let EQU be the curve of parameter pairs for which one critical point, say EQU, is preperiodic, with EQU lying on a periodic orbit of period EQU.
Here we assume that EQU is minimal and strictly positive.
Thus the curve gives maps such that one critical point maps to a fixed point of EQU, while in the quadrant EQU, gives maps such that one critical point maps to an orbit of period 2.
For further details, see Appendix A.
We can pass between the Cases EQU, EQU, EQU and EQU only by crossing over at least one of these curves.
In fact we need only the curves EQU and EQU in the half-plane EQU, as can be verified by study of Figure 1.
Similarly, we need only the curves EQU and EQU in the half-plane EQU.
Graphs of these four curves and the corresponding division of each parameter half-plane into four regions are shown in Figure 2, with irrelevant segments of the curves removed.
(A similar description of the Case boundaries can be given for the EQU parameter family consisting of suitably normalized polynomials of degree EQU.
There are analogous hypersurfaces EQU and EQU which separate the EQU regions EQU.
For EQU odd the description is very much like that in the cubic case, while for EQU even we need just three hypersurfaces, namely EQU and EQU in all cases, and also EQU when EQU.)
In the regions EQU and EQU of the cubic parameter plane there are many possibilities for complex behavior.
Some of the different kinds of behavior are distinguished in Figure 3.
In the region EQU we know that at least one of the two critical orbits must escape to infinity, but the other critical orbit may either escape (indicated by white in the figure), or remain bounded (indicated by light grey).
Similarly in Case EQU the two critical orbits may both behave chaotically (dark grey), or one or both may converge to attracting periodic orbits (lighter shades).
The regions EQU and EQU are colored white in this Figure, since they correspond to relatively dull dynamical behavior.
For a discussion of the methods used to make such figures, as well as their limitations, see Appendix C. rwhale_1152x900.ps pixels 1152 by 900 scaled 225 Figure 3.
Picture of the EQU parameter plane.
The boundaries between qualitatively different kinds of dynamic behavior have been indicated.
In the dark region, both critical orbits behave chaotically, while white indicates that both critical orbits escape to infinity, and intermediate shades indicate various intermediate forms of behavior.
(The illustrated region is the rectangle EQU.)
realAb.4.ps pixels 1152 by 900 scaled 225 Figure 4.
Corresponding picture in the EQU -plane.
(Region: EQU.)
For many purposes it is more natural to work in the EQU -parameter plane, where EQU.
The corresponding bifurcation diagram is shown in Figure 4.
Of course this figure incorporates only real cubics with positive leading coefficient.
For an analogous parametrization of cubics with negative leading coefficient we must work in the EQU -plane, where EQU so that EQU.
(See Figure 5.)
Figure 4 can be described roughly as the “double” of the upper half-plane in Figure 3, and Figure 5 as the double of the lower half-plane.
realAib.4.ps pixels 800 by 600 scaled 225 Figure 5.
Corresponding picture in the EQU -plane.
(Region: EQU.)
Inspection of magnified portions of Figure 3, 4 or 5 shows that several characteristic patterns are repeated many times on different scales.
Noteworthy are “swallow” shaped regions (Figure 7), “arch” shaped regions (Figure 11), and “product”-like regions (Figure 13).
We can partially explain these regions in terms of the dynamics of the associated maps EQU as follows.
A smooth map EQU with one or more critical points is said to be renormalizable if there exists a neighborhood EQU of the set of critical points so that: each component of EQU contains at least one critical point, the first return map EQU from EQU to itself is defined and smooth, and the union EQU has at least two distinct components.
(Here condition says that for each EQU there exists an integer EQU with EQU, and that the smallest such integer EQU is constant on each connected component of EQU.
Condition says that we exclude the trivial case where EQU is connected and maps into itself.)
More explicitly, a real cubic map EQU with (not necessarily distinct) real critical points is renormalizable if and only if it fits into one of the following four classes.
(See Figure 6.)
Case EQU.
Adjacent Critical Points.
There is an open interval EQU containing both critical points and an integer EQU so that the intervals EQU are pairwise disjoint, but EQU.
Case EQU.
There exist disjoint intervals EQU and EQU about the two critical points so that the first return map from the union EQU to itself is defined and smooth, interchanging these two components.
In other words, EQU and EQU for some EQU and EQU.
We will see that a universal model for this behavior occurs in a “biquadratic” map, that is, the composition of two quadratic maps.
Case EQU.
There are neighborhoods EQU and EQU as above, but the first return map carries both EQU and EQU into EQU.