Abstract
We define a self-similar set as the (unique) invariant set of an iterated function system of certain contracting affine functions. A topology on them is obtained (essentially) by inducing theC 1-topology of the function space. We prove that the measure function is upper semi-continuous and give examples of discontinuities. We also show that the dimension is not upper semicontinuous. We exhibit a class of examples of self-similar sets of positive measure containing an open set. IfC 1 andC 2 are two self-similar setsC 1 andC 2 such that the sum of their dimensionsd(C 1)+d(C 2) is greater than one, it is known that the measure of the intersection setC 2−C 1 has positive measure for almost all self-similar sets. We prove that there are open sets of self-similar sets such thatC 2−C 1 has arbitrarily small measure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. J. Falconer,The Geometry of Fractal Sets, Cambridge University Press, 1985.
M. J. Feigenbaum,Presentation Functions, Fixed Points, and a theory of Scaling Functions, J. Stat. Phys. Vol. 52, No 3-4, 527–569, 1989.
D. Hacon, N. C. Saldanha, J. J. P. Veerman,Some Remarks on Self-Affine Tilings, J. Exp. Math. Vol. 3, No. 4, 317–327, 1995.
J. E. Hutchinson,Fractals and Self-Similarity, Indiana Univ. Math. J., 30, No 5, 713–747, 1981.
R. Kenyon,Self-Similar Tilings, Ph. D. Thesis, Princeton University, 1990.
A. N. Kolmogorov, S. V. Fomin,Introductory Real Analysis, Dover, 1970.
J. Lagarias, Y. Wang,Tiling the Line with One Tile, Preprint, Georgia, 1994.
G. Moreira, Talk at the Conference of Dynamical Systems, held in Montevideo, april 1995.
A. Manning, H. McCluskey,Hausdorff Dimension for Horseshoes, Erg. Th. and Dyn. Sys. 3, 251–261, 1983.
J. M. Marstrand,Some Fundamental Properties of Plane Sets, Proc. Lond. Math. Soc. 4, 257–302, 1954.
A. M. Odlyzko,Non-Negative Digit Sets in Positional Number Systems, Proc. London Math. Soc., 3rd series, 37, 213–229, 1978.
J. Palis, Talk at the Conference of Dynamical Systems, held in Montevideo, april 1995.
J. Palis, F. Takens,Hyperbolicity and the Creation of Homoclinic Orbits, Ann. of Math. 82, 337–374, 1987.
J. Palis, F. Takens,Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, 1993.
J. Palis, J. C. Yoccoz,Homoclinic Tangencies for Hyperbolic Sets of large Hausdorff Dimensions, To appear.
A. Sannami,An Example of a Regular Cantor Set whose Difference is a Cantor Set with Positive Measure, Hokkaido Math. J. vol. 21, 7–24, 1992.
D. SullivanDifferential Structures on Fractal like Sets determined by Intrinsic Scaling Functions on Dual Cantor Sets, Nonlinear Evolution and Chaotic Phenomena, pg 101–110, NATO ASI series, Plenum, 1988.
F. Takens,Limit Capacity and Hausdorff Dimension of Dynamically Defined Sets, Lecture Notes in Math. 1331, 197–211, 211.
Author information
Authors and Affiliations
About this article
Cite this article
Veerman, J.J.P. Intersecting self-similar Cantor sets. Bol. Soc. Bras. Mat 26, 167–181 (1995). https://doi.org/10.1007/BF01236992
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01236992