  The formation and large-scale propagation of Poynting-dominated jets produced
by accreting, rapidly rotating black hole systems are studied by numerically
integrating the general relativistic magnetohydrodynamic equations of motion to
follow the self-consistent interaction between accretion disks and black holes.
This study extends previous similar work by studying jets till $t\approx
10^4GM/c^3$ out to $r\approx 10^4GM/c^2$, by which the jet is super- fast
magnetosonic and moves at a lab-frame bulk Lorentz factor of $\Gamma\sim 10$
with a maximum terminal Lorentz factor of $\Gamma_\infty\lesssim 10^3$. The
radial structure of the Poynting-dominated jet is piece-wise self-similar, and
fits to flow quantities along the field line are provided. Beyond the \alf
surface at $r\sim 10$--$100GM/c^2$, the jet becomes marginally unstable to (at
least) current-driven instabilities. Such instabilities drive shocks in the jet
that limit the efficiency of magnetic acceleration and collimation. These
instabilities also induce jet substructure with $3\lesssim\Gamma\lesssim 15$.
The jet is shown to only marginally satisfy the necessary and sufficient
conditions for kink instability, so this may explain how astrophysical jets can
extend to large distances without completely disrupting. At large distance, the
jet angular structure is Gaussian-like (or uniform within the core with sharp
exponential wings) with a half-opening angle of $\approx 5^\circ$ and there is
an extended component out to $\approx 27^\circ$. Unlike in some hydrodynamic
simulations, the environment is found to play a negligible role in jet
structure, acceleration, and collimation as long as the ambient pressure of the
surrounding medium is small compared to the magnetic pressure in the jet.
