10,174 research outputs found
Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure
For directed graph iterated function systems (IFSs) defined on R, we prove
that a class of 2-vertex directed graph IFSs have attractors that cannot be the
attractors of standard (1-vertex directed graph) IFSs, with or without
separation conditions. We also calculate their exact Hausdorff measure. Thus we
are able to identify a new class of attractors for which the exact Hausdorff
measure is known
Exact Hausdorff measure on the boundary of a Galton--Watson tree
A necessary and sufficient condition for the almost sure existence of an
absolutely continuous (with respect to the branching measure) exact Hausdorff
measure on the boundary of a Galton--Watson tree is obtained. In the case where
the absolutely continuous exact Hausdorff measure does not exist almost surely,
a criterion which classifies gauge functions according to whether
-Hausdorff measure of the boundary minus a certain exceptional set is
zero or infinity is given. Important examples are discussed in four additional
theorems. In particular, Hawkes's conjecture in 1981 is solved. Problems of
determining the exact local dimension of the branching measure at a typical
point of the boundary are also solved.Comment: Published at http://dx.doi.org/10.1214/009117906000000629 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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