Non-unique ergodicity, observers' topology and the dual algebraic lamination for $\R$-trees

TL;DR

This paper investigates the relationship between laminations, the observers' topology, and the metric structure of R-trees in free group actions, revealing that laminations determine the combinatorial structure but not the metric.

Contribution

It demonstrates that laminations dual to small free group actions uniquely determine the observers' topology but do not fix the R-tree's metric.

Findings

Laminations fully determine the combinatorial structure of R-trees.

The metric structure of R-trees is not uniquely determined by laminations.

Different metrics can share the same observers' topology.

Abstract

We continue in this article the study of laminations dual to very small actions of a free group F on R-trees. We prove that this lamination determines completely the combinatorial structure of the R-tree (the so-called observers' topology). On the contrary the metric is not determined by the lamination, and an R-tree may be equipped with different metrics which have the same observers' topology.