Abundance of Uenimodal Maps With Dense Critical Orbit and Prefixed Critical Orbit

Cao Yongluo; Wang Lanyu

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2000 > 21(4): 348-352

Cao Yongluo, Wang Lanyu. Abundance of Uenimodal Maps With Dense Critical Orbit and Prefixed Critical Orbit[J]. Applied Mathematics and Mechanics, 2000, 21(4): 348-352.

Citation:

Cao Yongluo, Wang Lanyu. Abundance of Uenimodal Maps With Dense Critical Orbit and Prefixed Critical Orbit[J]. Applied Mathematics and Mechanics, 2000, 21(4): 348-352.

Citation:

Cao Yongluo, Wang Lanyu. Abundance of Uenimodal Maps With Dense Critical Orbit and Prefixed Critical Orbit[J]. Applied Mathematics and Mechanics, 2000, 21(4): 348-352.

PDF( 331 KB)

Abundance of Uenimodal Maps With Dense Critical Orbit and Prefixed Critical Orbit

Cao Yongluo¹,
Wang Lanyu²

1.
Department of Mathematics, Suzhou University, Suzhou 215006, P R China;
2.
Institute of Mathematics, Academia Sinica, Beijing 100080, P R China

Received Date: 1998-11-20
Rev Recd Date: 1999-10-18
Publish Date: 2000-04-15

Abstract

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.
- unimodal map,
- quadratic family,
- invariant measure

FullText(HTML)

References(11)

References

[1]	Milnor J.On the concept of attractor[J].Comm Math Phys,1985,99(2):177～195.
[2]	Guckenheimer J.Sensitive dependence to initial conditions for one dimensional maps[J].Comm Math Phys,1979,70(1):133～160.
[3]	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.
[4]	Lyubich M,Milnor J.The unimodal Fibonacci map[J].J Amer Math Soc,1993,6(2):425～457.
[5]	Bruin H,Keller G,Nowicki T,et al.Wild Cantor attractors exist[J].Annals of Mathematics,1996,143(1):97～130.
[6]	Jakobson M V.Absolutely continuous invariant measures for one dimensional map[J].Comm Math Phys,1981,81(1):39～88.
[7]	Benedicks M,Carleson L.On iterations of 1-ax² on(-1,1)[J].Annals of Mathematics,1985,122(1):1～25.
[8]	Benedicks M,Carleson L.The dynamics of the Henon map[J].Annals of Mathematics,1991,133(1):73～169.
[9]	Mora L.Viana M.Abundance of strange attractors[J].Acta Math,1993,170(1):1～63.
[10]	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 Ⅰ Math,1992,315(1):69～72.
[11]	de Melo W,Van Strien S.One Dimensional Dynamics[M].Springer-Verlag,1993.