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ESCAPE QUARTERED THEOREM AND THE CONNECTIVITY OF THE JULIA SETS OF A FAMILY OF RATIONAL MAPS
Wang, Youming1,2; Yang, Fei1; Zhang, Song3; Liao, Liangwen1
2019-09-01
Source PublicationDISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
ISSN1078-0947
Volume39Issue:9Pages:5185-5206
AbstractIn this paper, we investigate the dynamics of the following family of rational maps f(lambda) (z) = z(2n) - 3(n+1) z(n) (z(2n) - lambda(n-1)) with one parameter lambda is an element of C* -{lambda:lambda(2n+2) =1}, where n >= 2. This family of rational maps is viewed as a singular perturbation of the bi- critical map P-n (z) = z(-n) if lambda not equal 0 is small. It is proved that the Julia set J (f(lambda)) is either a quasicircle, a Cantor set of circles, a Sierpinski carpet or a degenerate Sierpinski carpet provided the free critical orbits of f(lambda) are attracted by the super-attracting cycle 0 <-> 1. Furthermore, we prove that there exists suitable lambda such that J (f(lambda)) is a Cantor set of circles but the dynamics of f(lambda) on J (f(lambda)) is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of J (f(lambda)) is also proved if the free critical orbits are not attracted by the cycle 0 $ 1. Finally we give an estimate of the Hausdor ff dimension of the Julia set of f(lambda) in some special cases.
KeywordSingular perturbation Julia sets Cantor circles connectivity Hausdorff dimension
DOI10.3934/dcds.2019211
Language英语
Funding ProjectNSFC[11671092] ; NSFC[11671191] ; NSFC[11871208] ; Scientific Research Foundation of Hunan Provincial Education Department[16C0763] ; Natural Science Foundation of Hunan Province[2018JJ2159] ; Fundamental Research Funds for the Central Universities[0203-14380022] ; Fundamental Research Funds for the Central Universities[0203-14380025]
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000469023800011
PublisherAMER INST MATHEMATICAL SCIENCES-AIMS
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Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/34905
Collection中国科学院数学与系统科学研究院
Corresponding AuthorWang, Youming
Affiliation1.Nanjing Univ, Dept Math, Nanjing 210093, Jiangsu, Peoples R China
2.Hunan Agr Univ, Dept Appl Math, Changsha 410128, Hunan, Peoples R China
3.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China
Recommended Citation
GB/T 7714
Wang, Youming,Yang, Fei,Zhang, Song,et al. ESCAPE QUARTERED THEOREM AND THE CONNECTIVITY OF THE JULIA SETS OF A FAMILY OF RATIONAL MAPS[J]. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2019,39(9):5185-5206.
APA Wang, Youming,Yang, Fei,Zhang, Song,&Liao, Liangwen.(2019).ESCAPE QUARTERED THEOREM AND THE CONNECTIVITY OF THE JULIA SETS OF A FAMILY OF RATIONAL MAPS.DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,39(9),5185-5206.
MLA Wang, Youming,et al."ESCAPE QUARTERED THEOREM AND THE CONNECTIVITY OF THE JULIA SETS OF A FAMILY OF RATIONAL MAPS".DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 39.9(2019):5185-5206.
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