We investigate the effect of the nonmonotonic features of the macroscopic magnetic field B(x) and the de Hoffmann-Teller frame electrostatic potential &PHgr;HT(x) on the electron distribution functions within collisionless, fast mode shocks. The signatures of electron distribution functions are explored by using Liouville's theorem in the adiabatic approximation to map model upstream and downstream boundary electron velocity distribution functions to regions inside model shocks with monotonic and nonmonotonic magnetic fields under the empirically motivated approximation that Δ&PHgr;HT∝ΔB. In the case of shocks with monotonically increasing magnetic fields, we show that there are no exclusion regions and that the electron distribution function at all pitch angles and hence the electron temperature increase can be explained by the reversible behavior of magnetized electrons in the shock macroscopic electric and magnetic fields. However, at shocks with nonmonotonic magnetic fields, there exist regions of inaccessibility which are outside the domain of the one-Dimensional (1-D), steady state Vlasov-Liouville (V-L) approach as defined by the upstream and downstream boundaries. Such regions, if occupied, may be filled by electrons scattered into these regions by waves, or perhaps by reversible processes such as the adiabatic convection of electrons into these regions of a curved bow shock, or by coherent nonadiabatic access. As a further test, the V-L method is employed to study, for the first time, the detailed signatures of full 3-D electron velocity distribution functions observed by the Wind spacecraft through the resolved layer of a supercritical, fast mode Earth bow shock crossing. We demonstrate that much of the complex structure of the observed electron distribution function within the shock layer can be explained by the motion of adiabatic electrons in the nonmonotonic shock macroscopic magnetic and electric fields. However, a significant portion of electron phase space appears to be the remnants of electron phase space holes. The mechanisms responsible for allowing electrons to gain access to the exclusion regions are not well understood and may have important implications on the thermodynamic properties of collisionless shocks. ¿ 2001 American Geophysical Union