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.
@article{bwmeta1.element.bwnjournal-article-fmv160i1p1bwm,
author = {Manuel Mor\'an},
title = {Dynamical boundary of a self-similar set},
journal = {Fundamenta Mathematicae},
volume = {159},
year = {1999},
pages = {1-14},
zbl = {0931.28006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv160i1p1bwm}
}
Morán, Manuel. Dynamical boundary of a self-similar set. Fundamenta Mathematicae, Tome 159 (1999) pp. 1-14. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv160i1p1bwm/
[00000] [1] C. Bandt and S. Graff, Self-similar sets 7. A characterization of self-similar fractals with positive Hausdorff dimension, Proc. Amer. Math. Soc. 114 (1992), 995-1001. | Zbl 0823.28003
[00001] [2] J. K. Falconer, Fractal Geometry, Wiley, Chichester, 1990.
[00002] [3] J. K. Falconer, Sub-self-similar sets, Trans. Amer. Math. Soc. 347 (1995), 3121-3129. | Zbl 0844.28005
[00003] [4] J. Feder, Fractals, Plenum Press, New York, 1988.
[00004] [5] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. | Zbl 0598.28011
[00005] [6] T. Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 420 (1990). | Zbl 0688.60065
[00006] [7] J. Kigami and M. Lapidus, Weyl's problem for the spectral distribution of Laplacians on P.C.F. self-similar sets, Commun. Math. Phys. 158 (1993), 93-125. | Zbl 0806.35130
[00007] [8] B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1977.
[00008] [9] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, 1995. | Zbl 0819.28004
[00009] [10] M. Morán, Multifractal components of multiplicative functions, preprint. | Zbl 0998.28007
[00010] [11] M. Morán and J. M. Rey, Singularity of self-similar measures with respect to Hausdorff measures, Trans. Amer. Math. Soc. 350 (1998), 2297-2310. | Zbl 0899.28002
[00011] [12] M. Morán and J. M. Rey, Geometry of self-similar measures, Ann. Acad. Sci. Fenn. Math. 22 (1997), 365-386. | Zbl 0890.28005
[00012] [13] P. A. P. Moran, Additive functions of intervals and Hausdorff measures, Proc. Cambridge Philos. Soc. 42 (1946), 15-23. | Zbl 0063.04088
[00013] [14] A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), 111-115. | Zbl 0807.28005
[00014] [15] A. Schief, Self-similar sets in complete metric spaces, ibid. 121 (1996), 481-489. | Zbl 0844.28004
[00015] [16] S. Stella, On Hausdorff dimension of recurrent net fractals, ibid. 116 (1992), 389-400. | Zbl 0769.28007
[00016] [17] C. Tricot, Two definitions of fractal dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57-74.