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Abstract We construct an unfolding path in Outer space which does not converge in the boundary, and instead it accumulates on the entire 1-simplex of projectivized length measures on a nongeometric arational$${\mathbb R}$$-treeT. We also show thatTadmits exactly two dual ergodic projective currents. This is the first nongeometric example of an arational tree that is neither uniquely ergodic nor uniquely ergometric. 

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice $$X$$. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We show that the neighborhood of a minimal set is minimal. We characterize uniquely minimal sets of $$X$$: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of $$X$$: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate the graph $$G$$ of minimal sets whose vertices are congruence classes of minimal sets of $$X$$ and whose edges connect vertices which can be represented by minimal sets that differ by exactly one vertex. We prove that G has exactly one infinite component, has infinitely many isolated vertices and has bounded components of arbitrarily large size. Finally, we show that all minimal sets, except one, are connected.