  We have performed a detailed analysis of orbital motion in the vicinity of a
nearly extremal Kerr black hole. For very rapidly rotating black holes (spin
a=J/M>0.9524M) we have found a class of very strong field eccentric orbits
whose angular momentum L_z increases with the orbit's inclination with respect
to the equatorial plane, while keeping latus rectum and eccentricity fixed.
This behavior is in contrast with Newtonian intuition, and is in fact opposite
to the "normal" behavior of black hole orbits. Such behavior was noted
previously for circular orbits; since it only applies to orbits very close to
the black hole, they were named "nearly horizon-skimming orbits". Our analysis
generalizes this result, mapping out the full generic (inclined and eccentric)
family of nearly horizon-skimming orbits. The earlier work on circular orbits
reported that, under gravitational radiation emission, nearly horizon-skimming
orbits tend to evolve to smaller orbit inclination, toward prograde equatorial
configuration. Normal orbits, by contrast, always demonstrate slowly growing
orbit inclination (orbits evolve toward the retrograde equatorial
configuration). Using up-to-date Teukolsky-fluxes, we have concluded that the
earlier result was incorrect: all circular orbits, including nearly
horizon-skimming ones, exhibit growing orbit inclination. Using kludge fluxes
based on a Post-Newtonian expansion corrected with fits to circular and to
equatorial Teukolsky-fluxes, we argue that the inclination grows also for
eccentric nearly horizon-skimming orbits. We also find that the inclination
change is, in any case, very small. As such, we conclude that these orbits are
not likely to have a clear and peculiar imprint on the gravitational waveforms
expected to be measured by the space-based detector LISA.
