## Institute for Mathematical Sciences

## Preprint ims96-5

** M. Lyubich**
* Dynamics of quadratic polynomials, III: Parapuzzle and SBR measures.*

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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