  In this paper we study a new metric for comparing Betti numbers functions in
bidimensional persistent homology, based on coherent matchings, i.e. families
of matchings that vary in a continuous way. We prove some new results about
this metric, including its stability. In particular, we show that the
computation of this distance is strongly related to suitable filtering
functions associated with lines of slope 1, so underlining the key role of
these lines in the study of bidimensional persistence. In order to prove these
results, we introduce and study the concepts of extended Pareto grid for a
normal filtering function as well as of transport of a matching. As a
by-product, we obtain a theoretical framework for managing the phenomenon of
monodromy in 2D persistent homology.
