Summary
We obtain a critical function for which the Hausdorff measure of a branching set generated by a simple Galton-Watson process is positive and finite.
References
Athreya, K.B., Ney, P.E.: Branching processes. Berlin, Heidelberg: Springer 1972
Besicovitch, A.S.: On the fundamental geometrical properties of linearly measurable plan sets of points. Math. Ann.98, 422–464 (1928)
Falconer, K.J.: The Geometry of Fractal Sets. Cambridge: Cambridge University Press 1985
Falconer, K.J.: Random fractals. Math. Proc. Camb. Phil. Soc.100, 559–582 (1986)
Falconer, K.J.: Cut set sums and tree processes. Proc. Amer. Math. Soc.101(2), 337–346 (1987)
Falconer, K.J.: Fractal Geometry. New York: Wiley 1990
Graf, S., Maudlin, R.D. Williams, S.C.: The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc.71(381) (1988)
Hawkes, J.: Trees generated by a simple branching process. J. London Math. Soc.24(2), 373–384 (1981)
Kahane, J.P., Peyrière, J.: Sur certaines martingales de B. Mandelbrot, Adv. Math.22, 131–145 (1976)
Liu, Q.: Sur quelques problèmes à propos des processus de branchement, des flots dans les réseaux et des mesures de Hausdorff associées. Thèse de doctorat de L'Université Paris 6, Laboratoire de Probabilités, Paris, 1993
Lyons, R.: Random walks and percolation on trees. Ann. Probab.18, 931–952 (1990)
Lyons, R., Pemantle, R.: Random walk in a random environment and first passage percolation on trees. Ann. Probab.20, 125–135 (1992)
Maudlin, R.D., Williams, S.C.: Random constructions, asymptotic geometric and topological properties. Trans. Amer. Math. Soc.295, 325–346 (1986)
Neveu, J.: Arbre et processus de Galton—Watson. Ann. Inst. Henri Poincaré22, 199–207 (1986)
Rogers, C.A.: Hausdorff Measures. Cambridge: Cambridge University Press 1970
Taylor, S.J.: The measure theory of random fractals. Math. Proc. Cambridge Philos. Soc.100, 383–406 (1986)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Liu, Q. The exact hausdorff dimension of a branching set. Probab. Th. Rel. Fields 104, 515–538 (1996). https://doi.org/10.1007/BF01198165
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01198165
Mathematics Subject Classification (1991)
- 60J80
- 60J85
- 28A78
- 28A80