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Abstrakty
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
1-14
Opis fizyczny
Daty
wydano
1999
otrzymano
1997-11-06
poprawiono
1998-08-10
Twórcy
autor
- Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland
Bibliografia
- [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.
- [2] J. K. Falconer, Fractal Geometry, Wiley, Chichester, 1990.
- [3] J. K. Falconer, Sub-self-similar sets, Trans. Amer. Math. Soc. 347 (1995), 3121-3129.
- [4] J. Feder, Fractals, Plenum Press, New York, 1988.
- [5] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747.
- [6] T. Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 420 (1990).
- [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.
- [8] B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1977.
- [9] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, 1995.
- [10] M. Morán, Multifractal components of multiplicative functions, preprint.
- [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.
- [12] M. Morán and J. M. Rey, Geometry of self-similar measures, Ann. Acad. Sci. Fenn. Math. 22 (1997), 365-386.
- [13] P. A. P. Moran, Additive functions of intervals and Hausdorff measures, Proc. Cambridge Philos. Soc. 42 (1946), 15-23.
- [14] A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), 111-115.
- [15] A. Schief, Self-similar sets in complete metric spaces, ibid. 121 (1996), 481-489.
- [16] S. Stella, On Hausdorff dimension of recurrent net fractals, ibid. 116 (1992), 389-400.
- [17] C. Tricot, Two definitions of fractal dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57-74.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-fmv160i1p1bwm