Hausdorff dimension in graph directed constructions

Mauldin, R. Daniel; Williams, S. C.

doi:10.1090/S0002-9947-1988-0961615-4

Hausdorff dimension in graph directed constructions
HTML articles powered by AMS MathViewer

by R. Daniel Mauldin and S. C. Williams PDF

Trans. Amer. Math. Soc. 309 (1988), 811-829 Request permission

Abstract:

We introduce the notion of geometric constructions in ${{\mathbf {R}}^m}$ governed by a directed graph $G$ and by similarity ratios which are labelled with the edges of this graph. For each such construction, we calculate a number $\alpha$ which is the Hausdorff dimension of the object constructed from a realization of the construction. The measure of the object with respect to ${\mathcal {H}^\alpha }$ is always positive and $\sigma$-finite. Whether the ${\mathcal {H}^\alpha }$-measure of the object is finite depends on the order structure of the strongly connected components of $G$. Some applications are given.

References

P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), 15–23. MR 14397, DOI 10.1017/s0305004100022684
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
Richard S. Ellis, Entropy, large deviations, and statistical mechanics, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 271, Springer-Verlag, New York, 1985. MR 793553, DOI 10.1007/978-1-4613-8533-2
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027
K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
Vladimir Drobot and John Turner, Hausdorff dimension and Perron-Frobenius theory, Illinois J. Math. 33 (1989), no. 1, 1–9. MR 974007
C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
Jacques Marion, Mesures de Hausdorff et théorie de Perron-Frobenius des matrices non-negatives, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 4, 99–125 (French, with English summary). MR 812320
Helmut Cajar, Billingsley dimension in probability spaces, Lecture Notes in Mathematics, vol. 892, Springer-Verlag, Berlin-New York, 1981. MR 654147
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