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

ℝ-trees and laminations for free groups II: the dual lamination of an ℝ-tree

“…It turns out that, contrary to L 2 ∞ (T ), the set L 1 (T ) depends not just on the topology but actually on the metric of T . This is discussed in detail in [5] and in a forthcoming paper by the authors.…”

“…The converse inclusion with respect to Proposition 5.3 does not hold in general, as will be seen in Section 7. In fact, one has to regard L 1 (T ) as a finer invariant of T than the algebraic lamination L 2 Ω (T ), which only depends on a weakened version of the topology of T , compare [5], while L 1 (T ) may change when different R-tree structures are varying on a given topological tree T . This will be detailed in a forthcoming paper by the authors.…”

“…On the other hand, we have seen above that the three recurrent laminary languages associated to these three F N -invariant sets are equal. The set L 1 ∞ (L 2 Ω (T )) deserves some further attention, since it depends not on the metric on T , but only on the observer's topology on T ; compare [5]. Contrary to what seems to be indicated by the results of [5], the set L 1 ∞ (L 2 Ω (T )) shows that the lamination L 2 (T ) alone suffices, without invoking the metric on T , to exhibit certain completion points of the topological tree T as lying 'far out at infinity'.…”

“…Note also that the lamination L 2 (T ) introduced in this paper has been successfully used in [5] to characterize the underlying topological structure of R-trees which stays invariant when the metric is F N -equivariantly changed (the so-called 'non-uniquely ergometric R-trees').…”

Pretrees and the shadow topology

“…In the preceding paper [39], Ward denoted by σ the same topology in the case of dendritic spaces. T. Coulbois, A. Hilion, and M. Lustig [12] described this topology for the case of R-trees, calling it "the observers' topology" and attributing the invention of this term to V. Guirardel. P. de la Harpe and J.-P. Préaux, following N. Monod and Y. Shalom, described a version of the same topology for the case of the space that is the union of a tree and its ends, and referred to it as to "the shadow topology".…”