## 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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