Graphs of continuous functions and packing dimension

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• 作者：Jia Liu^a ; ^{liujia860319@163.com" class="auth_mail" title="E-mail the corresponding author} ; Bo Tan^b ; ^{tanbo@mail.hust.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Jun Wu^b ; ^{jun.wu@mail.hust.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Packing dimension ; Upper-box dimension ; Graph of function
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：435
• 期：2
• 页码：1099-1106
• 全文大小：279 K

文摘

We prove that for a typical continuous function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15010367&_mathId=si1.gif&_user=111111111&_pii=S0022247X15010367&_rdoc=1&_issn=0022247X&md5=0ab307b096eec0782b7355172c92cfd1" title="Click to view the MathML source">f∈C(X)class="mathContainer hidden">class="mathCode">$f\in C(X)$ over an uncountable compact metric space X, the packing dimension of its graph is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15010367&_mathId=si3.gif&_user=111111111&_pii=S0022247X15010367&_rdoc=1&_issn=0022247X&md5=2f508a5543e034b7459149cc2cf769ba" title="Click to view the MathML source">dim_P⁡(X)+1class="mathContainer hidden">class="mathCode">${\dim }_P(X)+1$; we also consider decompositions of functions in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15010367&_mathId=si4.gif&_user=111111111&_pii=S0022247X15010367&_rdoc=1&_issn=0022247X&md5=8f093f0e909fd2d27463a989419637d9" title="Click to view the MathML source">C([0,1])class="mathContainer hidden">class="mathCode">$C([0,1])$ in terms of upper box dimension as well as packing dimension, which are quite different from the case of Hausdorff dimension.