Simultaneous multifractal analysis of the branching and visibility measure on a Galton-Watson tree

Adam Kinnison, Peter Morters

Research output: Contribution to journalArticle

Abstract

On the boundary of a Gallon Watson tree we can define the visibility measure by splitting mass equally between the children of each vertex, and the branching measure by splitting unit mass equally between all vertices in the nth generation and then letting n go to infinity. The multifractal structure of each of these measures is well studied. In this paper we address the question of a joint multifractal spectrum, i.e. we ask for the Hausdorff dimension of the boundary points which simultaneously have an unusual local dimension for both these measures. The resulting two-parameter spectrum exhibits a number of surprising new features, among them the emergence of a swallowtail-shaped spectrum for the visibility measure in the presence of a nontrivial condition on the branching measure.
Original languageEnglish
Pages (from-to)226-245
Number of pages20
JournalAdvances in Applied Probability
Volume42
Issue number1
DOIs
Publication statusPublished - Mar 2010

Keywords

  • two-dimensional multifractal analysis
  • two-parameter spectrum
  • branching process
  • mixed spectrum
  • percolation
  • multifractal spectrum
  • random tree
  • self-similar fractal

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