How many future measures can there be?

The question addressed in this paper is the worst-case growth rate, for ergodic processes, in the number of conditional measures on n-steps in the future, given the past, that are a fixed distance apart. It is shown that if error is measured using the variational (i.e. distributional) distance then doubly exponential growth is possible, while if error is measured using the (d) over bar -metric then more than exponential growth is possible. The question of whether doubly exponential growth is possible in the (d) over bar case is left open.