Coupled skinny Baker's maps and the Kaplan-Yorke conjecture (Q2852526)

\textit{J. L. Kaplan} and \textit{J. A. Yorke} [Lect. Notes Math. 730, 204--227 (1979; Zbl 0448.58020)] conjectured that the box-counting dimension of the attractor of a typical dynamical system coincides with its Lyapunov or Kaplan-Yorke dimension \(D_L\) defined using the Lyapunov exponents. In later work [\textit{J. D. Farmer} et al., in: Order in chaos, Proc. int. Conf., Los Alamos/N.M. 1982, Physica D 7, 153--180 (1983; Zbl 0561.58032)] and [\textit{P. Frederickson} et al., J. Differ. Equations 49, 185--207 (1983; Zbl 0515.34040)] this was refined, replacing the box-counting dimension with other dimensions of the physical measure on the attractor. The resulting conjectured typical equalities are together labelled the ``Kaplan-Yorke conjecture''. Here the conjectured equality between the information dimension and the Lyapunov dimension is studied, in a neigbourhood of a dynamical system comprising two uncoupled systems. The particular motivation for this work is to explore whether coupling the two uncoupled systems typically restores the equality between the two dimensions. For a class of systems built from so-called skinny baker's maps, it is shown that coupling in one of the two possible directions the dimension equality is restored for a prevalent set of coupling functions, while for coupling in the other direction the dimensions remain different for all coupling functions. The authors conjecture that the dimensions coincide prevalently for bi-directional coupling, and that the one-sided phenomena seen in these examples holds more generally for many uni-directionally coupled systems in higher dimensions.