## Institute for Mathematical Sciences

## Preprint ims97-8

** M. Lyubich**
* Almost Every Real Quadratic Map is Either Regular or Stochastic*

Abstract: We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it
that the set of infinitely renormalizable parameter values in
the real quadratic family $P_c: x\mapsto x^2+c$ has zero
measure. This yields the statement in the title (where
``regular'' means to have an attracting cycle and
``stochastic'' means to have an absolutely continuous
invariant measure). An application to the MLC problem is given.

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