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 in a certain class of piecewise linear Bernoulli toral linked twist maps, given any there is a Bernoulli toral linked twist map with positive Lyapunov exponents defined only on a set of measure zero such that is within of in the metric.