On the coordinate functions of Peano curves

Makarov, B.; Podkorytov, A.

doi:10.1090/spmj/1441

On the coordinate functions of Peano curves
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by B. M. Makarov and A. N. Podkorytov
Translated by: N. Tsilevich

St. Petersburg Math. J. 28 (2017), 115-125

DOI: https://doi.org/10.1090/spmj/1441

Published electronically: November 30, 2016

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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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