By Peano curve I mean a continuous map from the unit interval that fills the unit square in $\mathbb R^2$.

In the paper:

Shchepin, E. V.; Bauman, K. E., Minimal Peano curve, Proc. Steklov Inst. Math. 263, 236-256 (2008); translation from Tr. Mat. Inst. Steklova 263, 251-271 (2008). ZBL1201.37061.

the authors mention the square to linear ratio, defined for a Peano curve $p:[0,1]\rightarrow[0,1]\times [0,1]$, as: $$\sup_{t_1,t_1\in[0,1]} \frac{|p(t_1)-p(t_2)|^2}{|t_1-t_2|}.$$

They prove that there exists a unique, up to isometry, regular diagonal Peano curve of fractal genus 9 that maps a unit interval onto a unit square and whose square-to-linear ratio is less than 6.

My question is: Are there properties such as the square-to-linear ratio that can be used to classify Peano curves up to isometry or some weaker class of transformations?

Discrete properties that separate Peano curves into disjoint categories like whether they are self-crossing or not would also interesting.


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