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).