  Non-continuous "jumps" of Apparent Horizons occur generically in 3+1 (binary)
black hole evolutions. The dynamical trapping horizon framework suggests a
spacetime picture in which these "Apparent Horizon jumps" are understood as
spatial cuts of a single spacetime hypersurface foliated by (compact)
marginally outer trapped surfaces. We present here some work in progress which
makes use of uni-parametric sequences of (axisymmetric) binary black hole
initial data for exploring the plausibility of this spacetime picture. The
modelling of Einstein evolutions by sequences of initial data has proved to be
a successful methodological tool in other settings for the understanding of
certain qualitative features of evolutions in restricted physical regimes.

General Relativity and Quantum Cosmology