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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

On the coordinate functions of Peano curves


Authors: B. M. Makarov and A. N. Podkorytov
Translated by: N. Tsilevich
Original publication: Algebra i Analiz, tom 28 (2016), nomer 1.
Journal: St. Petersburg Math. J. 28 (2017), 115-125
MSC (2010): Primary 26A16; Secondary 28A12
DOI: https://doi.org/10.1090/spmj/1441
Published electronically: November 30, 2016
MathSciNet review: 3591069
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Abstract | References | Similar Articles | Additional Information

Abstract: A construction of “nonsymmetric” plane Peano curves is described whose coordinate functions satisfy the Lipschitz conditions of orders $\alpha$ and $1-\alpha$ for some $\alpha$. It is proved that these curves are metric isomorphisms between the interval $[0,1]$ and the square $[0,1]^2$. This fact is used to show that the graphs of their coordinate functions have the maximum possible Hausdorff dimension for a given smoothness.


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Additional Information

B. M. Makarov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia
Email: BM1092@gmail.com

A. N. Podkorytov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia
Email: a.podkorytov@gmail.com

Keywords: Peano curve, Lipschitz condition, Hausdorff dimension
Received by editor(s): September 7, 2015
Published electronically: November 30, 2016
Article copyright: © Copyright 2016 American Mathematical Society