The behavior of the electron distribution function within mirror mode waves in the Earth's magnetosheath is studied using observations made by the electron experiment on Active Magnetospheric Particle Tracer Explorers United Kingdom Subsatellite (AMPTE UKS). The results of a detailed pitch angle analysis show that in the mirror mode troughs the electron distribution displays two distinct features related to particle trapping: (1) The deeply trapped electrons (those with pitch angles close to 90¿ in the magnetic field troughs) are significantly cooled with respect to the rest of the electron population and (2) the shallowly trapped electrons (those with pitch angles just greater than the critical pitch angle for trapping) are significantly heated with respect to the rest of the electron population. This is explained as a result of the trapped electron population undergoing a combination of Fermi acceleration and deceleration and betatron deceleration. The untrapped electron population appears to behave adiabatically as it travels alternately through mirror mode peaks and troughs. This behavior is confirmed by the use of Liouville mapping techniques, which also allow us to estimate the electric potential associated with the mirror mode wave. ¿ 1998 American Geophysical Union