Polynomial-like maps
Nuria Fagella

A polynomial-like map of degree d is a holomorphic map (MathWorld) f: U' → U such that every point in U has exactly d preimages in U', where U, U' are open sets isomorphic to disc and U contains U' in its interior.

Let f : U' → U be a polynomial-like map. The filled Julia set Kf of f is defined as the set of points in U' that never leave U' under iteration.

The Straightening Theorem Let f : U' → U be a polynomial-like map of degree d. Then f is hybrid equivalent (quasi-conformally conjugate) to a polynomial P of degree d. Moreover, if Kf is connected, then P is unique up to (global) conjugation by an affine map.

In particular, a hybrid equivalence implies that corresponding Julia sets are homeomorphic (MW). This theorem explains why one finds copies of Julia sets of polynomials in the dynamical planes of all kinds of functions. If f is polynomial- like of degree two and its Julia set is connected then f is hybrid equivalent to a polynomial z2 + c for a unique value of c. This may also be true for other families of polynomial-like maps of degree large than two, as long as the resulting class of polynomials has a unique representative in each affine class. See [1] for details.

Example A The obvious example is an actual polynomial P of degree d, restricted to a large enough open set. You see quadratic Julia set J(-1) here. Open sets U, U' are enclosed by the Γ, Γ' equipotential curves and Γ = P(Γ'). The triple (P|U' , U', U) is a polynomial like map of degree two.

The theory of polynomial-like mappings of A.Douady and J.Hubbard explains why the very particular family of polynomials is important for the understanding of iteration of a much wider class of functions namely those that locally behave as polynomials do.

Example B.1 Consider the cubic polynomial Pa(z) = z3 - 3a2z - 2a3 - a. For all value of a, the critical point ω2 = -a is a fixed (superattracting) point. For a = - 0.6 the critical point ω1 = a escapes to infinity. The open set U is enclosed by the equipotential curve Γ. Then the preimage of Γ under P is a figure eight curve Γ'. This curve bounds two connected components U' and V. U' is the component that contains the critical point ω2 with a bounded orbit, U' maps to U with degree two, i.e. every point in U has exactly two preimages in U'.

By the Straightening theorem, P-0.6(z) restricted to the open set U' is hybrid equivalent to a quadratic polynomial and hence, to a polynomial of the form Qc = z2 + c. In this case c = 0, since Qo(z) iz the only quadratic polynomial of this form with the critical point being fixed. Note that only the largest component in U' corresponds to the filled Julia set of the polynomial-like map of degree 2. Let f is the restriction of polynomial P to a set U'. As V maps to U with degree one, hence, there are points in U' that map to V and come back to U' afterwards never leaving the set U. Such points do not belong to Kf since they are not in U' at all times but they belong to KP since they do not escape to infinity under iteration. Therefore KP might have more connected components than Kf but not larger ones.
Example B.2 For the polynomial Ra(z) = z3 - 3a2z + [(9a2 - 4)½ + a - 4a3]/2 the critical point ω2 = -a is a point of period 2 cycle for all a values. For a = -0.75 (the right image) the critical point ω1 = a escapes to infinity. R-0.75(z) restricted to U' as above, is hybrid equivalent to a quadratic polynomial Q-1(z) with the critical point being of period two (see Example A).

Example C Let f(z) = π cos(z) is an entire transcendental function, U' is the open simply connected domain
    U' = { |Im(z)| < 1.7,   |Re(z)+π| < 2 },
and U = f(U'). Since U' contains only one critical point ω = -π, it follows that f maps U' to U with degree two. Hence the triple {f|U', U', U} is a polynomial-like of degree two. Since the critical point -π is fixed under f, the largest component (around -π below) is homeomorphic to the filled Julia set of Qo(z) = z2.
As usual, the phenomena in dynamical plane are reflected in parameter space. To the right you see the parameter l-plane for the mapping
    fλ(z) = λ cos(z).
There is a copy of the Mandelbrot set with λ = π as the center of its main cardioid.

[1] Nuria Fagella The theory of polynomial-like mappings - The importance of quadratic polynomials

Contents     Previous: The Mandelbrot set renormalization   Next: Renormalization
updated 12 May 2002