Fractal dimensions for inclusion hyperspaces and non-additive measures
1) Kasimir the Great University in Bydgoszcz Institute of Mathematics, Bydgoszcz, Poland; 2) Vasyl’ Stefanyk Precarpathian National University Department of Mathematics and Computer Science 57 Shevchenka St., Ivano-Frankivsk, Ukraine
Analogues of Hausdorff dimension, upper and lower box dimensions for the inclusion hyperspaces and non-additive regular measures (capacities) on metric compacta are introduced. Their relations to the respective dimensions of sets and additive measures are investigated. Methods for finding and estimating fractal dimensions of self-similar inclusion hyperspaces and self-similar non-additive measures are presented.
Hausdorff dimension; box dimension; measure; capacity; self-similarity; iterated function system
1. M. Barnsley, J. Hutchinson, O. Stenflo, V-variable fractals and superfractals, preprint, 2003, arXiv.org: math/0312314.
2. G. Choquet, Theory of capacity, Ann. Institute Fourier, 5 (1953-1954), 131-295.
3. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, 2003.
4. J.E. Hutchinson, L. Rushendorf, Random fractal measures via the contraction method, Indiana Univ. Math. J., 47 (1998), 471-487.
5. Lin Zhou, Integral representation of continuous comonotonically additive functionals, Trans. Amer. Math. Soc., 350 (1998), ¹5, 1811-1822.
6. P. Mattila, M. Moran, J.-M. Rey, Dimension of a measure, Studia Math., 142 (2000), ¹3, 219-233.
7. O.R. Nykyforchyn, Fractal capacities and iterated function systems, Mathematical Gerold of Shevchenko Scientific Society, 5 (2008), 259-273.
8. Ya.B. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, The Chicago University Press, Chicago and London, 1997.
9. A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122 (1994), ¹1, 111-115.
10. D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica, 57 (1989), 571–587.
11. E.R. Vrscay, From fractal image compression to fractal-based methods in mathematics, in Fractals in Multimedia, ed. by M.F. Barnsley, D. Saupe and E.R. Vrscay, New York, Springer-Verlag, 2002.
12. M.M. Zarichnyi, O.R. Nykyforchyn, Capacity functor in the category of compacta, Sbornik: Mathematics, 199 (2008), ¹2, 159–184.
13. M. Zarichnyi, A. Teleiko, Categorical Topology of Compact Hausdorff Spaces, VNTL Publ., Lviv, 1999.
|Full text of paper|
|Table of content of issue|