Dynamical boundary of a self-similar set

Manuel Morán

Fundamenta Mathematicae (1999)

  • Volume: 160, Issue: 1, page 1-14
  • ISSN: 0016-2736

Abstract

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Given a self-similar set E generated by a finite system Ψ of contracting similitudes of a complete metric space X we analyze a separation condition for Ψ, which is obtained if, in the open set condition, the open subset of X is replaced with an open set in the topology of E as a metric subspace of X. We prove that such a condition, which we call the restricted open set condition, is equivalent to the strong open set condition. Using the dynamical properties of the forward shift, we find a canonical construction for the largest open set V satisfying the restricted open set condition. We show that the boundary of V in E, which we call the dynamical boundary of E, is made up of exceptional points from a topological and measure-theoretic point of view, and it exhibits some other boundary-like properties. Using properties of subself-similar sets, we find a method which allows us to obtain the Hausdorff and packing dimensions of the dynamical boundary and the overlapping set in the case when X is the n-dimensional Euclidean space and Ψ satisfies the open set condition. We show that, in this case, the dimension of these sets is strictly less than the dimension of the set E.

How to cite

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Morán, Manuel. "Dynamical boundary of a self-similar set." Fundamenta Mathematicae 160.1 (1999): 1-14. <http://eudml.org/doc/212378>.

@article{Morán1999,
abstract = {Given a self-similar set E generated by a finite system Ψ of contracting similitudes of a complete metric space X we analyze a separation condition for Ψ, which is obtained if, in the open set condition, the open subset of X is replaced with an open set in the topology of E as a metric subspace of X. We prove that such a condition, which we call the restricted open set condition, is equivalent to the strong open set condition. Using the dynamical properties of the forward shift, we find a canonical construction for the largest open set V satisfying the restricted open set condition. We show that the boundary of V in E, which we call the dynamical boundary of E, is made up of exceptional points from a topological and measure-theoretic point of view, and it exhibits some other boundary-like properties. Using properties of subself-similar sets, we find a method which allows us to obtain the Hausdorff and packing dimensions of the dynamical boundary and the overlapping set in the case when X is the n-dimensional Euclidean space and Ψ satisfies the open set condition. We show that, in this case, the dimension of these sets is strictly less than the dimension of the set E.},
author = {Morán, Manuel},
journal = {Fundamenta Mathematicae},
keywords = {self-similar sets; Hausdorff dimension; open set condition; packing dimension; separation condition; restricted open set condition; strong open set condition; dynamical boundary},
language = {eng},
number = {1},
pages = {1-14},
title = {Dynamical boundary of a self-similar set},
url = {http://eudml.org/doc/212378},
volume = {160},
year = {1999},
}

TY - JOUR
AU - Morán, Manuel
TI - Dynamical boundary of a self-similar set
JO - Fundamenta Mathematicae
PY - 1999
VL - 160
IS - 1
SP - 1
EP - 14
AB - Given a self-similar set E generated by a finite system Ψ of contracting similitudes of a complete metric space X we analyze a separation condition for Ψ, which is obtained if, in the open set condition, the open subset of X is replaced with an open set in the topology of E as a metric subspace of X. We prove that such a condition, which we call the restricted open set condition, is equivalent to the strong open set condition. Using the dynamical properties of the forward shift, we find a canonical construction for the largest open set V satisfying the restricted open set condition. We show that the boundary of V in E, which we call the dynamical boundary of E, is made up of exceptional points from a topological and measure-theoretic point of view, and it exhibits some other boundary-like properties. Using properties of subself-similar sets, we find a method which allows us to obtain the Hausdorff and packing dimensions of the dynamical boundary and the overlapping set in the case when X is the n-dimensional Euclidean space and Ψ satisfies the open set condition. We show that, in this case, the dimension of these sets is strictly less than the dimension of the set E.
LA - eng
KW - self-similar sets; Hausdorff dimension; open set condition; packing dimension; separation condition; restricted open set condition; strong open set condition; dynamical boundary
UR - http://eudml.org/doc/212378
ER -

References

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  12. [12] M. Morán and J. M. Rey, Geometry of self-similar measures, Ann. Acad. Sci. Fenn. Math. 22 (1997), 365-386. Zbl0890.28005
  13. [13] P. A. P. Moran, Additive functions of intervals and Hausdorff measures, Proc. Cambridge Philos. Soc. 42 (1946), 15-23. Zbl0063.04088
  14. [14] A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), 111-115. Zbl0807.28005
  15. [15] A. Schief, Self-similar sets in complete metric spaces, ibid. 121 (1996), 481-489. Zbl0844.28004
  16. [16] S. Stella, On Hausdorff dimension of recurrent net fractals, ibid. 116 (1992), 389-400. Zbl0769.28007
  17. [17] C. Tricot, Two definitions of fractal dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57-74. 

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