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
