Coupled skinny baker's maps and the Kaplan–Yorke conjecture

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. The particular subsystems we consider are skinny baker's maps and we consider uni-directional coupling. We are interested in whether coupling 'typically' restores the equality of the dimensions. For coupling in one of the possible directions, we prove that the dimensions coincide for a prevalent (measure-theoretically typical) 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 this phenomenon that the two dimensions differ robustly occurs more generally for many classes of uni-directionally coupled (skew-product) systems in higher dimensions. This is a joint work with Brian Hunt (University of Maryland, College Park).