  Many ambitious experiments have been proposed to constrain dark energy and
detect its evolution. At present, observational constraints are consistent with
a cosmological constant and there is no firm evidence for any evolution in the
dark energy equation of state w. In this paper, we pose the following question:
suppose that future dark energy surveys constrain w at low redshift to be
consistent with -1 to a percent level accuracy, what are the implications for
models of dynamical dark energy? We investigate this problem in a
model-independent way by following quintessence field trajectories in `energy'
phase-space. Attractor dynamics in this phase-space leads to two classes of
acceptable models: 1) models with flat potentials, i.e. an effective
cosmological constant, and 2) models with potentials that suddenly flatten with
a characteristic kink. The prospect of further constraining the second class of
models from distance measurements and fluctuation growth rates at low redshift
(z<3) seems poor. However, in some models of this second class, the dark energy
makes a significant contribution to the total energy density at high redshift.
Such models can be further constrained from observation of the cosmic microwave
background anisotropies and from primordial nucleosynthesis. It is possible,
therefore, to construct models in which the dark energy at high redshift causes
observable effects, even if future dark energy surveys constrain w at low
redshift to be consistent with -1 to high precision.
