On the arithmetic sum of middle-cantor sets

Authors

  • Eduardo Muñoz Universidad Católica del Norte.
  • Jaime Vera Universidad Católica del Norte.
  • Sergio Plaza Universidad de Santiago de Chile.

DOI:

https://doi.org/10.22199/S07160917.1995.0001.00005

Abstract

In this article we study the arithmetic sum (difference) set K? + K?  in I = [0, 1] in terms of the parameters (?, ?) ?  I x I, where K? and K? are middle-Cantor sets contained in I. We describe two regions, A and B, in the parameter space (?, ?)  where the characterization of the arithmetic sum set K? + K?  is given.

Author Biographies

Eduardo Muñoz, Universidad Católica del Norte.

Depto. de Matemáticas, Facultad de Ciencias.

Jaime Vera, Universidad Católica del Norte.

Depto. de Matemáticas, Facultad de Ciencias.

Sergio Plaza, Universidad de Santiago de Chile.

Depto. de Matemáticas, Facultad de Ciencias.

References

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[3] Falconer, K. The Geometry of Fractal Sets. Cambridge Univ. Press (1985).

[4] Mendes, P., Oliveira, F. On the topological structure of the arithmetic sum of two Cantor sets. Nonlinearity 7, 329-343 (1994).

[5] Newhouse, S. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. Math. !.H. E. S. 50, 101-151 (1979).

[6] Palis, J. Homoclinic orbits, hyperbolic dynamics and dimension of Cantor sets. Contemporary Mathematics vol. 58, part III, 204-216 (1987).

[7] Palis, J., Takens, F. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations: fractal dimensions and infinitely many attractors. Cambridge Univ. Press (1993).

[8] Sannami, A. An example of regular Cantor set whose difference is a Cantor set with positive Lebesgue measure. Hokkaido Math. Journal, vol. XXI {1), 7-24 (1992).

[9] Williams, R. F. How big is the intersection of two Cantor sets?. Contemporary Mathematics, vol. 117, 163-175 (1991).

Published

2018-04-03

How to Cite

[1]
E. Muñoz, J. Vera, and S. Plaza, “On the arithmetic sum of middle-cantor sets”, Proyecciones (Antofagasta, On line), vol. 14, no. 1, pp. 51-63, Apr. 2018.

Issue

Section

Artículos