Skip to main content Accessibility help

Dimensions associated with recurrent self-similar sets

  • Anca Deliu (a1), J. S. Geronimo (a1), R. Shonkwiler (a1) and D. Hardin (a2)


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.



Hide All
[1]Barnsley, M. F., Elton, J. and Hardin, D.. Recurrent iterated function systems. Constr. Approx. 5 (1989), 331.
[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), 12181242.
[3]Barnsley, M. F. and Harrington, A. N.. The calculus of fractal interpolating functions. J. Approx. Theory 57 (1989), 1434.
[4]Bedford, T.. Crinkly curves, Markov partitions, and dimension. Ph. Thesis, Warwick University (1984).
[5]Bedford, T.. Dimension and dynamics for fractal recurrent sets. J. London Math. Soc. 33 (1986), 89100.
[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).
[7]Bedford, T. and Urbanski, M.. The Box and Hausdorff Dimensions of Self-affine Sets. Delft University report 88–80 (1988).
[8]Bohr, T. and Rand, D.. The entropy function for characteristic exponents. Physica 25D (1987), 387393.
[9]Cawley, R. and Mauldin, R. D.. Multifractal decompositions of Moran fractals. Preprint.
[10]Falconer, K. J.Dimensions and measures of quasi-self-similar sets. Proc. Amer. Math. Soc. 106 (1989), 543554.
[11]Falconer, K. J.. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103 (1988), 339350.
[12]Feller, W.. An Introduction to Probability Theory and Its Applications (John Wiley and Sons, 1950).
[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), 8998.
[14]Hutchinson, J.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713747.
[15]Mandelbrot, B.. The Fractal Geometry of Nature (W. H. Freeman and Co., 1982).
[16]McMullen, C.. The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96 (1984), 19.
[17]Young, L.-S.. Dimension, entropy and Lyapunov exponents. Ergodic Theory Dynamical Systems 2 (1982), 109129.
[18]Tricot, C.. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 (1982), 5774.
[19]Walters, P.. An Introduction to Ergodic Theory (Springer-Verlag, 1982).

Dimensions associated with recurrent self-similar sets

  • Anca Deliu (a1), J. S. Geronimo (a1), R. Shonkwiler (a1) and D. Hardin (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed