  We examine the evolution of extremal spherically symmetric black holes,
developing both general theory as well as the specific cases of (charged) null
dust and massless scalar field spacetimes. As matter accretes onto extremal
marginally trapped tubes, they generically evolve to become non-extremal with
the initial extremal horizon bifurcating into inner and outer non-extremal
horizons. At the start of this process arbitrarily slow matter accretion can
cause a geometrically invariant measure of horizon growth to jump from zero to
infinity. We also consider dynamical horizons that are extremal throughout
their evolution and see that such spacetimes contain two extremal black hole
horizons: an inner isolated one and an outer dynamical one. We compare these
extremal dynamical horizons with the dynamical extreme event horizon spacetimes
of Murata, Reall and Tanahashi.
