2014/07/25 by Ulrich Oertel, Oertel, Ulrich
Computer Science · Engineering · Mathematics · #57M99 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #Digital Filter Design and Implementation #FOS: Mathematics #Geometric Topology (math.GT) #Polynomial and algebraic computation #Probability (math.PR) #math.CA #math.GT #math.PR #msc:57M99
paper · pdf · doi:10.48550/arxiv.1407.7066
48 pages, 12 figures. This version contains extensive corrections, changes, additional material. The title was changed
openalex publication_date 2014/07/25 · arxiv created 2016/07/27 · arxiv updated 2016/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We describe a construction of ordered algebraic structures (ordered abelian semigroups, ordered commutative semirings, etc.) and describe applications to codimension-1 laminations. For a suitable ordered semi- algebraic structure \mathbb L and measurable space X we define \mathbb L-measures ν on X. If L is a codimension-1 lamination in a manifold, it often admits transverse \mathbb L-measures for some \mathbb L. Transverse \mathbb L-measures can be used to understand classes of laminations much larger than the class of laminations admitting transverse positive \mathbb R-measures. In particular, we show that "finite or infinite depth measured laminations" are laminations admitting transverse measures with values in a certain ordered semiring \mathbb O satisfying the additional property that locally the values lie in a smaller semiring \mathbb P. We consider the "realization problem:" In one version, this deals with the problem whether an \mathbb P-invariant weight vector assigned to a branched manifold B (satisfying certain branch equations) determines a lamination L carried by B with a transverse \mathbb O-measure inducing the weights on B. We describe further laminations which may not be \mathbb L-measured, but are "well-covered" by laminations with transverse \mathbb L-measures. We also investigate actions on \mathbb L-trees which are associated to essential laminations with transverse \mathbb L-measures. In appendices, we develop ideas about \mathbb L-measures a little further, for example showing that a \mathbb P-measure can be interpreted as a kind of probability measure.