A Bernoulli toral linked twist map without positive Lyapunov exponents

The presence of positive Lyapunov exponents in a dynamical system is often taken to be equivalent to the chaotic behavior of that system. We construct a Bernoulli toral linked twist map which has positive Lyapunov exponents and local stable and unstable manifolds defined only on a set of measure zero. This is a deterministic dynamical system with the strongest stochastic property, yet it has positive Lyapunov exponents only on a set of measure zero. In fact we show that for any map $g$ in a certain class of piecewise linear Bernoulli toral linked twist maps, given any $\epsilon >0$ there is a Bernoulli toral linked twist map $g'$ with positive Lyapunov exponents defined only on a set of measure zero such that $g'$ is within $\epsilon$ of $g$ in the $\bar{d}$ metric.