Abstract
For the so-called quadratic family, it is proved that, there exists a positive Lebesgue measure set E in the parameter space such that the corresponding map has a dense critical orbit in the support set of the invariant measure for almost all parameters in E; it is also proved that there is a dense set in E such that the critical orbit of the corresponding map enter the reversed fixed point.
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References
Milnor J. On the concept of attractor[J].Comm Math Phys, 1985,99(2):177–195.
Guckenheimer J. Sensitive dependence to initial conditions for one dimensional maps[J].Comm Math Phys, 1970,70(1):133–160
Blokh A M, Lyubich M Yu. Measurable dynamics of S-unimodal maps of the interval[J].Ann Sci École Norm Sup(4), 1991,24(4):545–573.
Lyubich M, Milnor J. The unimodal Fibonacci map[J].J Amer Math Soc, 1993,6(2):425–457.
Bruin H, Keller G, Nowicki T, et al. Wild Cantor attractors exist[J].Annals of Mathematics, 1996,143(1):97–130.
Jakobson M V. Absolutely continuous invariant measures for one dimensional map[J].Comm Math Phys, 1981,81(1):39–88.
Benedicks M, Carleson L. On iterations of 1−ax 2 on (−1,1)[J].Annals of Mathematics, 1985,122(1):1–25.
Benedicks M, Carleson L. The dynamics of the Henon map[J].Annals of Mathematics, 1991,133 (1):73–169.
Mora L. Viana M. Abundance of strange attractors[J].Acta Math, 1993,170(1):1–63.
Thieullen Ph, Tresser C, Young L S. Positive Liapunov exponents for generic one parameter families of unimodal maps[J].C R Acad Sci Paris Sér IMath, 1992,315(1):69–72.
de Melo W, Van Strien S.One Dimensional Dynamics[M]. Springer-Verlag, 1993.
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Foundation item: the National Natural Science Foundation of China (19501030)
Biography: Cao Yongluo (1967-)
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Yongluo, C., Lanyu, W. Abundance of unimodal maps with dense critical orbit and prefixed critica orbit. Appl Math Mech 21, 389–394 (2000). https://doi.org/10.1007/BF02463759
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DOI: https://doi.org/10.1007/BF02463759