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International Mathematics Research Notices (2000) 2000:333-351, doi:10.1155/S1073792800000192
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Copyright © 2000 Hindawi Publishing Corporation. All rights reserved.

Symbolic dynamics and Collet-Eckmann conditions

Stanislav Smirnov

We prove that unicritical polynomials with metrically generic combinatorics of the critical orbit satisfy the Collet-Eckmann conditions. Here metrically generic means except for a set of Hausdorff dimension zero, and combinatorics can be understood in either of the following senses: Markov partition itineraries, kneading sequences, external angles, or harmonic measure on the Mandelbrot set.

Particularly, except for a set of Hausdorff dimension zero of angles, all external rays for the degree d Mandelbrot set land at parameters c such that polynomial zd + c is Collet-Eckmann. This statement is in a sense the best possible, and is much stronger than saying that almost every c with respect to harmonic measure corresponds to a Collet-Eckmann polynomial.

Some of the theorems can be generalized to polynomials with many critical points and rational functions.


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