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| The Dimension Paradox in Parameter Space of Cosine Family. |
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Citation： |
Xiaojie HUANG.The Dimension Paradox in Parameter Space of Cosine Family.[J].Chinese Annals of Mathematics B,2020,41(4):645~656 |
| Page view： 578
Net amount： 452 |
Authors： |
Xiaojie HUANG; |
Foundation： |
This work was supported by the National Natural Science Foundation of China (Nos. 11671091, 11731003, 11771090), the Natural Science Foundation of Shanghai (No. 17ZR1402900) and the Science and Technology Foundation of Jiangxi Education Department (No. GJJ180944). |
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| Abstract： |
It is proved in this paper that the union of escaping parameter rays without endpoints for the cosine family Sk(z) = ek (ez + e-z ), where κ ∈ C is a parameter, has Hausdorff dimension 1, which implies that the ray endpoints alone have Hausdorff dimension 2. This shows that Karpi′nska’s dimension paradox occurs also in the parameter plane of the cosine family. |
Keywords： |
Dimension paradox, Hausdorff dimension, Parameter rays, Dynamic rays |
Classification： |
37F35, 37F10, 37F45 |
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