Transport maps, non-branching sets of geodesics and measure rigidity

In this paper we investigate the relationship between a general existence of transport maps of optimal couplings with absolutely continuous first marginal and the property of the background measure called essentially non-branching introduced by Rajala-Sturm (Calc.Var.PDE 2014). In particular, it is shown that the qualitative non-degenericity condition introduced by Cavalletti-Huesmann (Ann. Inst. H. Poincar\'e Anal. Non Lin\`eaire 2015) implies that any essentially non-branching metric measure space has a unique transport maps whenever initial measure is absolutely continuous. This generalizes a recently obtained result by Cavalletti-Mondino (Commun. Contemp. Math. 2017) on essentially non-branching spaces with the measure contraction condition $\mathsf{MCP}(K,N)$. In the end we prove a measure rigidity result showing that any two essentially non-branching, qualitatively non-degenerate measures on a fixed metric spaces must be mutually absolutely continuous. This result was obtained under stronger conditions by Cavalletti-Mondino (Adv.Math. 2016). It applies, in particular, to metric measure spaces with generalized finite dimensional Ricci curvature bounded from below.