# 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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Applications conservant une mesure absolument continue par rapport àdx sur [0, 1]
Sufficient conditions are given such that a differentiable, non invertible, mapg:[0, 1]↦[0, 1] leaves invariant a measure absolutely continuous with respect to the Lebesgue measure. In particular,