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Dynamics of quadratic polynomials, III: Parapuzzle and SBR measures

Mikhail Lyubich
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Abstract

This is a continuation of notes on dynamics of quadratic polynomials. In this part we transfer the our prior geometric result to the parameter plane. To any parameter value c in the Mandelbrot set (which lies outside of the main cardioid and little Mandelbrot sets attached to it) we associate a ``principal nest of parapuzzle pieces'' and show that the moduli of the annuli grow at least linearly. The main motivation for this work was to prove the following: Theorem B (joint with Martens and Nowicki). Lebesgue almost every real quadratic polynomial which is non-hyperbolic and at most finitely renormalizable has a finite absolutely continuous invariant measure.

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Mikhail Lyubich. 1996-06-15. Dynamics of quadratic polynomials, III: Parapuzzle and SBR measures. https://arxiv.org/abs/math/9606219

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