  We examine the nature of phase transitions occurring in strongly correlated
Fermi systems at the quantum critical point (QCP) associated with a divergent
effective mass. Conventional scenarios for the QCP involving collective degrees
of freedom are shown to have serious shortcomings. Working within the original
Landau quasiparticle picture, we propose an alternative topological scenario
for the QCP, in systems that obey standard Fermi liquid (FL) theory in advance
of the QCP. Applying the technique of Poincar\'e mapping, we analyze the
sequence of iterative maps generated by the Landau equation for the
single-particle spectrum at zero temperature. It is demonstrated that the Fermi
surface is subject to rearrangement beyond the QCP. If the sequence of maps
converges, a multi-connected Fermi surface is formed. If it fails to converge,
the Fermi surface swells into a volume that provides a measure of entropy
associated with formation of an exceptional state of the system characterized
by partial occupation of single-particle states and dispersion of their
spectrum proportional to temperature. Based on this dual scenario, the
thermodynamics of Fermi systems beyond the QCP exhibits striking departures
from the predictions of standard FL theory. Mechanisms for the release of the
entropy excess of the exceptional state are discussed.
