Denjoy-Wolff theorems, Hilbert metric nonexpansive maps and reproduction-decimation operators

Brian Lins, Roger Nussbaum

Research output: Contribution to journalArticle

10 Scopus citations

Abstract

Let K be a closed cone with nonempty interior in a Banach space X. Suppose that f : int K → int K is order-preserving and homogeneous of degree one. Let q : K → [0, ∞) be a continuous, homogeneous of degree one map such that q (x) > 0 for all x ∈ K {set minus} {0}. Let T (x) = f (x) / q (f (x)). We give conditions on the cone K and the map f which imply that there is a convex subset of ∂K which contains the omega limit set ω (x ; T) for every x ∈ int K. We show that these conditions are satisfied by reproduction-decimation operators. We also prove that ω (x ; T) ⊂ ∂ K for a class of operator-valued means.

Original languageEnglish (US)
Pages (from-to)2365-2386
Number of pages22
JournalJournal of Functional Analysis
Volume254
Issue number9
DOIs
StatePublished - May 1 2008

All Science Journal Classification (ASJC) codes

  • Analysis

Keywords

  • Denjoy-Wolff theorems
  • Diffusion on fractals
  • Dirichlet forms
  • Hilbert metric
  • Operator means
  • Positive operators

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