Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-19T03:51:50.070Z Has data issue: false hasContentIssue false
  • English
  • Français

Dimensions associated with recurrent self-similar sets

Published online by Cambridge University Press:  24 October 2008

Anca Deliu ,
J. S. Geronimo ,
R. Shonkwiler  and
D. Hardin
Anca Deliu
Affiliation:
Georgia Institute of Technology, School of Mathematics, Atlanta, Georgia 30332, U.S.A.
J. S. Geronimo
Affiliation:
Georgia Institute of Technology, School of Mathematics, Atlanta, Georgia 30332, U.S.A.
R. Shonkwiler
Affiliation:
Georgia Institute of Technology, School of Mathematics, Atlanta, Georgia 30332, U.S.A.
D. Hardin
Affiliation:
Vanderbilt University, Department of Mathematics, Nashville, Tennessee 37235, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

The Hausdorff and box dimensions for measures associated with recurrent self-similar sets generated by similitudes is explicitly given. The box dimension of the attractor associated with a class of two-dimensional affine maps is also computed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 2 , September 1991 , pp. 327 - 336
Copyright
Copyright © Cambridge Philosophical Society 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Barnsley, M. F., Elton, J. and Hardin, D.. Recurrent iterated function systems. Constr. Approx. 5 (1989), 3–31.CrossRefGoogle Scholar
[2]Barnsley, M. F.Elton, J. H., Hardin, D. P. and Massopust, P. R.. Hidden-variable fractal interpolation functions. SIAM J. Math. Anal. 20 (1989), 1218–1242.CrossRefGoogle Scholar
[3]Barnsley, M. F. and Harrington, A. N.. The calculus of fractal interpolating functions. J. Approx. Theory 57 (1989), 14–34.CrossRefGoogle Scholar
[4]Bedford, T.. Crinkly curves, Markov partitions, and dimension. Ph. Thesis, Warwick University (1984).Google Scholar
[5]Bedford, T.. Dimension and dynamics for fractal recurrent sets. J. London Math. Soc. 33 (1986), 89–100.CrossRefGoogle Scholar
[6]Bedford, T.. Hausdorff dimension and box dimension in self-similar sets. In Proc. Conf. Topology and Measure V (Binz, GDR, 1987), Wissenschaftliche Beiträge der Ernst-Moritz-Arndt Universität, Griefswald (1988).Google Scholar
[7]Bedford, T. and Urbanski, M.. The Box and Hausdorff Dimensions of Self-affine Sets. Delft University report 88–80 (1988).Google Scholar
[8]Bohr, T. and Rand, D.. The entropy function for characteristic exponents. Physica 25D (1987), 387–393.Google Scholar
[9]Cawley, R. and Mauldin, R. D.. Multifractal decompositions of Moran fractals. Preprint.Google Scholar
[10]Falconer, K. J.Dimensions and measures of quasi-self-similar sets. Proc. Amer. Math. Soc. 106 (1989), 543–554.CrossRefGoogle Scholar
[11]Falconer, K. J.. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103 (1988), 339–350.CrossRefGoogle Scholar
[12]Feller, W.. An Introduction to Probability Theory and Its Applications (John Wiley and Sons, 1950).Google Scholar
[13]Geronimo, J. S. and Hakdin, D. P.. An exact formula for the measure dimensions associated with a class of piecewise linear maps. Constr. Approx. 5 (1989), 89–98.CrossRefGoogle Scholar
[14]Hutchinson, J.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[15]Mandelbrot, B.. The Fractal Geometry of Nature (W. H. Freeman and Co., 1982).Google Scholar
[16]McMullen, C.. The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96 (1984), 1–9.CrossRefGoogle Scholar
[17]Young, L.-S.. Dimension, entropy and Lyapunov exponents. Ergodic Theory Dynamical Systems 2 (1982), 109–129.CrossRefGoogle Scholar
[18]Tricot, C.. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74.CrossRefGoogle Scholar
[19]Walters, P.. An Introduction to Ergodic Theory (Springer-Verlag, 1982).CrossRefGoogle Scholar