Email this article Peano-type curves, Liouville numbers, and microscopic sets
- Authors: Agadzhanov А.N.1
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Affiliations:
- Institute of Control Sciences of the Russian Academy of Sciences
- Issue: Vol 485, No 1 (2019)
- Pages: 7-10
- Section: Mathematics
- URL: https://journals.eco-vector.com/0869-5652/article/view/12801
- DOI: https://doi.org/10.31857/S0869-565248417-10
- ID: 12801
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Abstract
Peano-type curves in multidimensional Euclidean space are considered in terms of number theory. In contrast to curves constructed by D. Hilbert, H. Lebesgue, V. Sierpinski, and others, this paper presents results showing that each such curve is a continuous image of universal (shared by all curves) nowhere dense perfect subsets of the interval [0, 1] with a zero s-dimensional Hausdorff measure that consist of only Liouville numbers. An example of a problem in which a pair of continuous functions controlling the behavior of an oscillating system generates a Peano-type curve in the plane is given.
About the authors
А. N. Agadzhanov
Institute of Control Sciences of the Russian Academy of Sciences
Author for correspondence.
Email: ashot_ran@mail.ru
Russian Federation, 65, Profsoyuznaya Street, Moscow, 117997
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