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Topological invariance of the Collet-Eckmann property for $$S$$-unimodal maps. (English) Zbl 0898.58014
Two metric properties of $$S$$-unimodal maps which have been studied since the 1980s are the existence of an absolutely continuous invariant measure (a.c.i.m.) and the Collet-Eckmann condition. A question appeared, asked apparently independently by several people (Misiurewicz, Guckenheimer, Van Strien), of whether any of these conditions might be a topological invariant. Chances of that happening seemed dim, because the topological conjugacy cannot be expected to be absolutely continuous, and Hölder continuous is not enough since conjugacies to the tent map are Hölder continuous. The invariance of the Collet-Eckmann condition would follow if the conjugacy were shown to be quasi-symmetric, but this, though likely true, is not easy to prove. Indeed, it was shown that the existence of an a.c.i.m. is not a topological invariant.
The current paper proves that the Collet Eckmann condition is preserved under topological conjugacy. The proof presents a summary of the deep and difficult work done on the Collet-Eckmann condition. The second author had noticed a transparently topological condition which follows from Collet-Eckmann. Interestingly, that observation was made in the context of holomorphic dynamics. One notes that the holomorphic and $$S$$-unimodal cases are technically close because many proofs are based on Koebe’s distortion lemma, which is classical for univalent maps and has an exact counterpart for ones with positive Schwarzian derivative. The topological condition in turn easily implies that Lyapunov exponents of all repelling periodic orbits are bounded from below by a positive constant. The theorem now follows from a recent difficult result by Nowicki and Sands which states that for $$S$$-unimodal maps, uniformly positive Lyapunov exponents on periodic orbits imply the Collet-Eckmann condition.
It should finally be noted that the topological invariance persists even if the order of the critical point is not fixed; the critical point just needs to be of order greater than 1 and finite. In holomorphic dynamics, the question of whether uniformly positive Lyapunov exponents on periodic orbits imply the Collet-Eckmann condition remains open.

MSC:
 37E99 Low-dimensional dynamical systems 54H20 Topological dynamics (MSC2010) 37D99 Dynamical systems with hyperbolic behavior 37B99 Topological dynamics 26A18 Iteration of real functions in one variable 30C55 General theory of univalent and multivalent functions of one complex variable 37F99 Dynamical systems over complex numbers
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