Absolutely continuous invariant measures for one-parameter families of one-dimensional maps

@article{Jakobson1981AbsolutelyCI,
  title={Absolutely continuous invariant measures for one-parameter families of one-dimensional maps},
  author={Michael Jakobson},
  journal={Communications in Mathematical Physics},
  year={1981},
  volume={81},
  pages={39-88}
}
  • M. Jakobson
  • Published 1 September 1981
  • Mathematics
  • Communications in Mathematical Physics
Given a one-parameter familyfλ(x) of maps of the interval [0, 1], we consider the set of parameter values λ for whichfλ has an invariant measure absolutely continuous with respect to Lebesgue measure. We show that this set has positive measure, for two classes of maps: i)fλ(x)=λf(x) where 0<λ≦4 andf(x) is a functionC3-near the quadratic mapx(1−x), and ii)fλ(x)=λf(x) (mod 1) wheref isC3,f(0)=f(1)=0 andf has a unique nondegenerate critical point in [0, 1]. 
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