A partial description of parameter space of rational maps of degree two. I. (English) Zbl 0774.58035

Rees, Mary

Acta Math. 168, No. 1-2, 11-87 (1992).

This paper is an attempt to classify hyperbolic rational maps of degree 2. The reader finds here an extensive introduction to Part I and to Part II of this paper [IMS Preprint 1991/4, Stony Brook, New York, 1991; per revr.]. For polynomials of degree 2 with connected Julia set there is Thurston’s description: Suppose the critical point 0 for \(f_ c(z)=z^ 2+c\) is periodic. One finds a lamination of the unit disc \(\mathbb{D}\) with the leaves being intervals with the end points in \(\partial\mathbb{D}\) invariant under a continuous map \(s_ c\) which is \(z\mapsto z^ 2\) in \(\hat\mathbb{C}\backslash\mathbb{D}\). Collapsing leaves (and finite-sided polygons with edges being leaves) to points, one obtains a branched cover homotopic and conjugate to \(f_ c\), and the image of \(\text{cl}\mathbb{D}\) after contraction corresponds to the filled-in Julia set for \(f_ c\). Minor leaves of these laminations give a parameter lamination which, after taking the closure and contracting leaves to points, gives the Mandelbrot set \(M\) (assuming the conjecture that \(M\) is locally connected is true). The author extends this model to describe hyperbolic rational functions of degree 2. Every such critically finite map \(f\) (i.e. with one critical point, say \(c_ 1\), periodic of period \(m\geq 2\), the other, \(c_ 2\), also with finite forward orbit \(O(c_ 2))\) falls into one of the following classes: (II) \(c_ 2\in O(c_ 1)\backslash\{c_ 1\}\), (III) \(c_ 2\in\bigcup_{n\geq 0} f^{-n}(O(c_ 1))\backslash O(c_ 1)\), (IV) \(c_ 2\) is periodic and \(O(c_ 1)\neq O(c_ 2)\) (class (I) is reserved for polynomials with \(f^ n_ c(0)\to\infty)\).
The polynomial-and-path theorem asserts that each branched cover \(f\) in (III) can be obtained up to homotopy as \(\sigma_ \gamma\circ P\), where \(P\) is a polynomial, \(\gamma:[0,1]\to\hat\mathbb{C}\) is a curve with \(\gamma(0)=\infty\), \(\gamma(1)\in O(c_ 1)\backslash\{c_ 1\}\) and \(\sigma_ \gamma\) is a homeomorphism \(h_ 1\) which is the identity outside a neighbourhood of \(\gamma\), maps \(\gamma(0)\) to \(\gamma(1)\) and is isotopic via \(h_ t\) to the identity \(h_ 0\) so that \(h_ t(\gamma(0))=\gamma(t)\). Types II and IV are of the form \(\sigma_ \eta\circ\sigma_ \gamma\circ P\) (for these types the author assumes that \(f\) is rational).
Having a lamination in \(\mathbb{D}\) related to a polynomial \(P\) one cuts \(\partial D\) at the ends of the leaves of the full orbit of the minor leaf or at the vertices of the full orbit of the adjacent finite-sided gap. What remains in \(\partial\mathbb{D}\) is a Cantor set \(K\). Then one can complete (rebuild) the lamination with a lamination coming from outside \(\text{cl} \mathbb{D}\) into the gaps. Thus one obtains a lamination invariant for a branched cover map homotopic and semi-conjugate to \(f\), up to a holomorphic change of coordinates (lamination map equivalence and conjugacy theorems).
Given a branched cover \(f\) of one of the types II-IV preserving the kind of lamination described above, to decide whether there is a conformal structure invariant for \(f\) the author uses Thurston’s criterion in a version simplified for degree 2: the nonexistence of S. Levy’s cycles. Type IV corresponds to (and generalizes) the mating construction. Types II and III generalize B. Wittner’s captures.
Given a polynomial \(P\) with \(c_ 1=0\) periodic of period \(m\geq 2\), the author studies the one-parameter family (parametrised by the second critical point \(c_ 2\), or curves \(\gamma)\): Minor leaves of the above laminations are defined and they, together with boundary leaves of minor gaps, give “parameter lamination” for branched covers of degree 2 born from \(P\), “admissible” leaves corresponding to rational maps, extending Thurston’s minor leaves lamination.
In the admissible boundary theorem (proved mostly in Part II, extending Lei’s matability theorem [T. Lei, Ergodic Theory Dyn. Syst. 12, No. 3, 589-620 (1992; Zbl 0756.58024)] it is proved that each leaf in the boundary of the set of admissible leaves is either a side of a minor gap of an invariant lamination or a limit of such sides, with period of the second branching point \(\leq m\) and Levy’s cycles of period \(m\).
The “rabbit”-born examples: \(P(z)=(z-a)(z-1)/z^ 2\), \(a\neq 0\), where \(0\mapsto\infty\mapsto 1\mapsto 0\), are discussed.
Related new papers are by T. Lei [op. cit.], the author [Part II, op. cit.; Invent. Math. 100, No. 2, 357-382 (1990; Zbl 0712.30022)] and in Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. II, 1295-1304 (1991; Zbl 0742.58032)] and J. Milnor [“Remarks on quadratic rational maps”, IMS Preprint 1992/14, Stony Brook, New Yor, 1992; per revr.].

MSC:

37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30D05	Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable

Keywords:

Mandelbrot set

Citations:

Zbl 0756.58024; Zbl 0712.30022; Zbl 0742.58032

References:

[1]	[A]Ahlfors, L.,Lectures on Quasiconformal Mappings. D. van Nostrand Co., 1966.
[2]	[B]Brolin, M., Invariant sets under iteration of rational functions.Ark. Mat., 6 (1966), 103–144. · Zbl 0127.03401 · doi:10.1007/BF02591353
[3]	[D]Daouady, A., Systèmes dynamiques holomorphes.,Séminaire Bourbaki, 1983.Astérique, 105–106 (1983), 39–63.
[4]	[D-H1]Douady, A. & Hubbard, J. H., Etudes dynamiques des polynômes complexes, avec la collaboration de P. Lavaurs, Tan Lei, P. Sentenac. Parts I and II. Publications Mathématiques d’Orsay, 1985.
[5]	[D-H2]–, Itérations des polynômes quadratiques complexes.C. R. Acad. Sci. Paris Sér. I, 294 (1982), 123–126. · Zbl 0483.30014
[6]	[D-H3]Douady, A. & Hubbard, J. H., A proof of thruston’s topological characterization of rational functions. Mittag-Leffler preprint, 1985.
[7]	[Du]Duren, P.,Univalent Functions Springer, New York, 1983.
[8]	[F]Fatou, P., Mémoire sur les équations fonctionelles.Bull. Soc. Math. France, 47 (1919), 161–271; 48 (1920), 33–96 and 208–314. · JFM 47.0921.02
[9]	[F-L-P]Fahti, A., Laudenbach, F. & Poénaru, V., Traaux de Thurston sur les surfaces.Astérisque, 66–67 (1979).
[10]	[Fr]Franks, J., Anosov diffeomorphisms.Proc. Sympos. Pure Math., 14 (1968), 61–93.
[11]	[J]Julia, G., Itérations des applications fonctionelles.J. Math. Pures Appl., 8 (1918), 47–245.
[12]	[La]Lavaurs, P., Une déscription combinatoire de l’involution défnie parM sur les rationnels à dénominateur impair.C. R. Acad. Sci. Paris Sér. I, 303 (1986), 143–146.
[13]	[L]Levy, S. V. F., Critically finite rational maps. Thesis, Princeton University, 1985.
[14]	[M]Magnus, W., Über Automorphismen von Fundementalengruppen berandeter Flachen.Math. Ann., 109 (1934), 617–646. · Zbl 0009.03901 · doi:10.1007/BF01449158
[15]	[McM]McMullen, C., Automorphisms of rational maps, inHolomorphic Functions and Moduli, Vol. I. Math. Sci. Res. Inst. Publ., no. 10, pp. 31–60 (Berkeley, 1986). Springer, New York-Berlin, 1988.
[16]	[M-S-S]Mané, R., Sad, P &Sullivan, D., On the dynamics of rational maps.Ann. Sci. Ecole Norm. Sup. (4), 16 (1983), 193–217. · Zbl 0524.58025
[17]	[M-T]Milnor, J. W. & Thurston, W. P., On iterated maps of the interval. Preprint, 1981. · Zbl 0664.58015
[18]	[R]Rees, M., Components of degree two hyperbolic rational maps.Invent. Math., 100 (1990), 357–382. · Zbl 0712.30022 · doi:10.1007/BF01231191
[19]	[S]Shishikura, M., To appear.
[20]	[T]Thurston, W. P., On the combinatorics of iterated rational maps. Preprint, Princeton University and I.A.S., 1985.
[21]	[T2]The geometry and topology of three-manifolds. Notes, Princeton Universisty, 1978.
[22]	[TL]Tan Lei, Accouplements des polynômes complexes. Thèse, Université de Paris-Sud, Orsay, 1987.
[23]	[W]Wittner, B., On the bifurcation loci of rational maps of degree two. Thesis, Cornell University, 1988.