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

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Abstract

Given a one-parameter familyf λ(x) of maps of the interval [0, 1], we consider the set of parameter values λ for whichf λ 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)f λ(x)=λf(x) where 0<λ≦4 andf(x) is a functionC 3-near the quadratic mapx(1−x), and ii)f λ(x)=λf(x) (mod 1) wheref isC 3,f(0)=f(1)=0 andf has a unique nondegenerate critical point in [0, 1].

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  1. Central Scientific-Research Economic Institute, Smolenskii Boulevard, G-117, Moscow, USSR

    M. V. Jakobson

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  1. M. V. Jakobson
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Communicated by Ya. G. Sinai

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Jakobson, M.V. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun.Math. Phys. 81, 39–88 (1981). https://doi.org/10.1007/BF01941800

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  • DOI: https://doi.org/10.1007/BF01941800

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