A Self-Similar Dendrite with One-Point Intersection and Infinite Post-Critical Set

Year 2018, Volume: 2 Issue: 2, 70 - 73, 30.06.2018

Andrei Tetenov , Prabhjot Singh

https://doi.org/10.31197/atnaa.417783

Abstract

We build an example of a system S of similarities in R^2 whose attractor is a plane dendrite $K\supset [0,1]$ which satisfies one point intersection property, while the post-critical set of the system S is a countable set  whose natural projection to K is dense in the middle-third Cantor set.

Keywords

self-similar set, attractor, post-critically finite, open set condition, dendrite

References

• C. Bandt, N.V. Hung and H. Rao, On the open set condition for self-similar fractals, Proc. Amer. Math. Soc. 134 (2005), 1369-1374
• Charatonik J., Charatonik W., Dendrites, Aportaciones Mat. Comun. 22 (1998), 227-253.
• Hutchinson, J.E. Fractals and self-similarity. Indiana Univ. Math. J 30, 1981, 713-747
• Kigami, J. Analysis on Fractals. Cambridge University Press, 2001
• P.A.P. Moran, Additive functions of intervals and Hausdor measure Proc. Cambridge Philos. Soc. 42 (1946), 15-23