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
Full-text PDF
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.
- G. Peano, Sur une courbe, qui remplit toute une aire plane, Math. Ann. 36 (1890), no. 1, 157–160 (French). MR 1510617, DOI https://doi.org/10.1007/BF01199438
- N. N. Luzin, The theory of functions of a real variable, Uchpedgiz, Moscow, 1940. (Russian)
- Hans Sagan, Space-filling curves, Universitext, Springer-Verlag, New York, 1994. MR 1299533
- A. S. Besiovitch and H. D. Ursell, Sets of fractional dimensions (V): On dimensional numbers of some continuous curves, J. London Math. Soc. 12 (1937), 18–25.
- S. A. Kline, On curves of fractional dimensions, J. London Math. Soc. 20 (1945), 79–86. MR 16452, DOI https://doi.org/10.1112/jlms/s1-20.2.79
- K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
- H. Steinhaus, La courbe de Peano et les fonctions indépendantes, C. R. Acad. Sci. Paris 202 (1936), 1961–1963.
- Adriano M. Garsia, Combinatorial inequalities and smoothness of functions, Bull. Amer. Math. Soc. 82 (1976), no. 2, 157–170. MR 582776, DOI https://doi.org/10.1090/S0002-9904-1976-13975-4
- E. R. Love and L. C. Young, Sur une classe de fonctionnelles linéaires, Fund. Math. 28 (1937), 243–257.
- John A. R. Holbrook, Stochastic independence and space-filling curves, Amer. Math. Monthly 88 (1981), no. 6, 426–432. MR 622959, DOI https://doi.org/10.2307/2321827
- Boris Makarov and Anatolii Podkorytov, Real analysis: measures, integrals and applications, Universitext, Springer, London, 2013. Translated from the 2011 Russian original. MR 3089088
Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 26A16, 28A12
Retrieve articles in all journals with MSC (2010): 26A16, 28A12
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