  In this paper we characterize the Nazarov-Sklyanin hierarchy for the
classical periodic Benjamin-Ono equation in two complementary degenerations:
for the multi-phase initial data (the periodic multi-solitons) at fixed
dispersion and for bounded initial data in the limit of small dispersion.
First, we express this hierarchy in terms of a piecewise-linear function of an
auxiliary real variable which we call a dispersive action profile and whose
regions of slope $\pm 1$ we call gaps and bands, respectively. Our expression
uses Kerov's theory of profiles and Kre\u{\i}n's spectral shift functions.
Next, for multi-phase initial data, we identify Baker-Akhiezer functions in
Dobrokhotov-Krichever and Nazarov-Sklyanin and prove that multi-phase
dispersive action profiles have finitely-many gaps determined by the
singularities of their Dobrokhotov-Krichever spectral curves. Finally, for
bounded initial data independent of the coefficient of dispersion, we show that
in the small dispersion limit, the dispersive action profile concentrates
weakly on a convex profile which encodes the conserved quantities of the
dispersionless equation. To establish the weak limit, we reformulate
Szeg\H{o}'s first theorem for Toeplitz operators using spectral shift
functions. To illustrate our results, we identify the dispersive action profile
of sinusoidal initial data with a profile found by Nekrasov-Pestun-Shatashvili
and its small dispersion limit with the convex profile found by Vershik-Kerov
and Logan-Shepp.
