Absolutely continuous invariant measures for one-parameter families of one-dimensional maps |
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Authors: | M V Jakobson |
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Institution: | (1) Central Scientific-Research Economic Institute, Smolenskii Boulevard, G-117 Moscow, USSR |
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Abstract: | Given a one-parameter familyf
(x) of maps of the interval 0, 1], we consider the set of parameter values for whichf
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
(x)=f(x) where 0<4 andf(x) is a functionC
3-near the quadratic mapx(1–x), and ii)f
(x)=f(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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Keywords: | |
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