Monatshefte für Mathematik

, Volume 180, Issue 4, pp 693–712 | Cite as

Peano curves with smooth footprints

Article
  • 122 Downloads

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.

Keywords

Space-filling curves Smoothness Convexity Line fields 

Mathematics Subject Classification

26A30 26E10 

References

  1. 1.
    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)Google Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    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)Google Scholar
  4. 4.
    Falconer, K.: Fractal geometry: mathematical foundations and applications, 2nd edn. Wiley (2003)Google Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    Pach, J., Rogers, C.: Partly convex Peano curves. Bull. Lond. Math. Soc. 15(4), 321–328 (1983)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Sagan, H.: Space-Filling Curves. Universitext. Springer, New York (1994)Google Scholar
  8. 8.
    Úbeda García, J.I.: Aspectos geométricos y topológicos de la curvas\(\alpha \) -densas. PhD thesis, Universidad de Alicante (2006)Google Scholar
  9. 9.
    Vince, A., Pach, J., Rogers, C.A.: E3139 (large discs in convex unions). Am. Math. Mon. 95(8), 765–767 (1988)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Vince, A., Wilson, D.C.: A convexity preserving Peano curve. Houston J. Math. 12(2), 295–304 (1986)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Departamento de MatemáticaPontifícia Universidade Católica do Rio de JaneiroRio de JaneiroBrazil

Personalised recommendations