Dimensions and measures of quasi self-similar sets

Falconer, K. J.

doi:10.1090/S0002-9939-1989-0969315-8

Dimensions and measures of quasi self-similar sets
HTML articles powered by AMS MathViewer

by K. J. Falconer PDF

Proc. Amer. Math. Soc. 106 (1989), 543-554 Request permission

Abstract:

We show that sets with certain quasi self-similar properties have equal Hausdorff and box-packing dimensions and also have positive and finite Hausdorff measure at the dimensional value. A number of applications of these results to particular examples are given.

References

Tim Bedford, Dimension and dynamics for fractal recurrent sets, J. London Math. Soc. (2) 33 (1986), no. 1, 89–100. MR 829390, DOI 10.1112/jlms/s2-33.1.89
Rufus Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11–25. MR 556580
K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
Benoit B. Mandelbrot, The fractal geometry of nature, Schriftenreihe für den Referenten. [Series for the Referee], W. H. Freeman and Co., San Francisco, Calif., 1982. MR 665254
John McLaughlin, A note on Hausdorff measures of quasi-self-similar sets, Proc. Amer. Math. Soc. 100 (1987), no. 1, 183–186. MR 883425, DOI 10.1090/S0002-9939-1987-0883425-3
Claude Tricot Jr., Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 1, 57–74. MR 633256, DOI 10.1017/S0305004100059119
David Rand, Universality and renormalisation in dynamical systems, New directions in dynamical systems, London Math. Soc. Lecture Note Ser., vol. 127, Cambridge Univ. Press, Cambridge, 1988, pp. 1–56. MR 953969
David Ruelle, Bowen’s formula for the Hausdorff dimension of self-similar sets, Scaling and self-similarity in physics (Bures-sur-Yvette, 1981/1982) Progr. Phys., vol. 7, Birkhäuser Boston, Boston, MA, 1983, pp. 351–358. MR 733478