  Using direct numerical simulation we study the behavior of the maximal
Lyapunov exponent in thin-layer turbulence, where one dimension of the system
is constrained geometrically. Such systems are known to exhibit transitions
from fully three dimensional turbulence through a mixed two and three
dimensional phenomenology state and then onto fully two dimensional dynamics.
We find a discontinuous jump in the Lyapunov exponent at this second transition
implying the predictability of such systems can change dramatically. Such
transitions are seen in a number of different turbulent systems, for example
those undergoing strong rotation, hence these results may be relevant for the
predictability of complicated real world flows. The Lyapunov exponent is found
to provide a particularly clear measure of the transition to two dimensional
dynamics. Finally, the application of these results to atmospheric
predictability with regard to high-resolution modeling is examined.
