Almost Every Real Quadratic Map Is Either Regular or Stochastic

@inproceedings{Lyubich1997AlmostER,
  title={Almost Every Real Quadratic Map Is Either Regular or Stochastic},
  author={Mikhail Lyubich},
  year={1997}
}
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 → 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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