Can we ever distinguish between quintessence and a cosmological constant?

Many ambitious experiments have been proposed to constrain dark energy and detect its evolution. At the 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 less than or similar to 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.