  The Kaplan-Yorke conjecture states that for "typical" dynamical systems with
a physical measure, the information dimension and the Lyapunov dimension
coincide. We explore this conjecture in a neighborhood of a system for which
the two dimensions do not coincide because the system consists of two uncoupled
subsystems. We are interested in whether coupling "typically" restores the
equality of the dimensions. The particular subsystems we consider are skinny
baker's maps, and we consider uni-directional coupling. For coupling in one of
the possible directions, we prove that the dimensions coincide for a prevalent
set of coupling functions, but for coupling in the other direction we show that
the dimensions remain unequal for all coupling functions. We conjecture that
the dimensions prevalently coincide for bi-directional coupling. On the other
hand, we conjecture that the phenomenon we observe for a particular class of
systems with uni-directional coupling, where the information and Lyapunov
dimensions differ robustly, occurs more generally for many classes of
uni-directionally coupled systems (also called skew-product systems) in higher
dimensions.
