Abstract
We construct Peano curves \(\gamma : [0,\infty ) \rightarrow \mathbb {R}^2\) whose “footprints” \(\gamma ([0,t])\), \(t>0\), have \(C^\infty \) boundaries and are tangent to a common continuous line field on the punctured plane \(\mathbb {R}^2 {\backslash }\{\gamma (0)\}\). Moreover, these boundaries can be taken \(C^\infty \)-close to any prescribed smooth family of nested smooth Jordan curves contracting to a point.
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Notes
Actually, Pach and Rogers obtained a curve with the additional properties that \(\gamma ([0,1]) = [0,1]^2\) and \(\gamma ([t,1])\) is convex for each t. These properties can easily be obtained by modifying our curve: the argument goes exactly as in the last paragraph of §3 in [6].
References
Bonatti, C., Franks, J.: A Hölder continuous vector field tangent to many foliations. In: Modern dynamical systems and applications, pp. 299–306. Cambridge University Press, Cambridge (2004)
Choquet, G.: Lectures on analysis. In: Marsden, J., Lance T., Gelbart, S. (eds.) Integration and Topological Vector Spaces, vol. 1, W. A. Benjamin, Inc., New York-Amsterdam (1969)
Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved Problems in Geometry: Unsolved Problems in Intuitive Mathematics. Problem Books in Mathematics, vol. 2. Springer, New York (1994) (Corrected reprint of the 1991 original)
Falconer, K.: Fractal geometry: mathematical foundations and applications, 2nd edn. Wiley (2003)
Milet, P.H.: Curvas de Peano e Campos de Direções [Peano Curves and Line Fields]. Master’s thesis, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro (2011)
Pach, J., Rogers, C.: Partly convex Peano curves. Bull. Lond. Math. Soc. 15(4), 321–328 (1983)
Sagan, H.: Space-Filling Curves. Universitext. Springer, New York (1994)
Úbeda García, J.I.: Aspectos geométricos y topológicos de la curvas\(\alpha \) -densas. PhD thesis, Universidad de Alicante (2006)
Vince, A., Pach, J., Rogers, C.A.: E3139 (large discs in convex unions). Am. Math. Mon. 95(8), 765–767 (1988)
Vince, A., Wilson, D.C.: A convexity preserving Peano curve. Houston J. Math. 12(2), 295–304 (1986)
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Communicated by A. Constantin.
The first named author was partially supported by project Fondecyt 1140202 and project Anillo ACT1103 (Chile). The second named author was supported by FAPERJ (Brazil).
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Bochi, J., Milet, P.H. Peano curves with smooth footprints. Monatsh Math 180, 693–712 (2016). https://doi.org/10.1007/s00605-016-0899-8
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DOI: https://doi.org/10.1007/s00605-016-0899-8