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Rigidity for real polynomials. (English) Zbl 1129.37020

The paper presents a proof of the topological (or combinatorial) rigidity property for real polynomials with all critical points real and nondegenerate, which completes the last step in solving the density of Axiom A conjecture in real one-dimensional dynamics.
It is a long standing open problem whether Axiom A (hyperbolic) maps are dense in reasonable families of one-dimensional dynamical systems. The paper gives a proof of the following:
Density of Axiom A Theorem. Let \(f\) be a real polynomial of degree \(d\geq 2\). Assume that all critical points of \(f\) are real and that \(f\) has a connected Julia set. Then \(f\) can be approximated by hyperbolic real polynomials of degree \(d\) with real critical points and connected Julia sets.
A polynomial is called hyperbolic if all of its critical points are contained in the basin of an attracting cycle or infinity. The method used in the paper shows that the theorem is still true without assumption on connectedness of a Julia set: Any real polynomial \(f\) with all critical points real can be approximated by hyperbolic real polynomials with the same degree and with real critical points.
It is announced that in the sequel to the paper it will be shown that Axiom A maps on the real line are dense in the \(C^k\) topology for \(k=1,2,\dots,\infty,\omega\).
The proof of the theorem is through the quasi-symmetric rigidity approach due to D. Sullivan [Am. Math. Soc. Centen. Publ. 2, 417–466 (1992; Zbl 0936.37016)].
The Density of Axiom A Theorem follows from the Rigidity Theorem stated below. For any positive integer \(d\geq 2\), let \(\mathcal F_d\) denote the family of polynomials \(f\) of degree \(d\) with following properties:
the coefficient of \(f\) are all real;
\(f\) has only real critical points which are all nondegenerate;
\(f\) does not have any neutral periodic point;
the Julia set of \(f\) is connected.
Rigidity Theorem. Let \(f\) and \(\tilde f\) be two polynomials in \(\mathcal F_d\). If they are topologically conjugate as dynamical systems on the real line \(\mathbb R\), then they are quasiconformally conjugate as dynamical systems on the complex plane \(\mathbb C\).

MSC:

37E05 Dynamical systems involving maps of the interval
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37E20 Universality and renormalization of dynamical systems
37F25 Renormalization of holomorphic dynamical systems

Citations:

Zbl 0936.37016
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