<p>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 <i><b>g</i></b> in a certain class of piecewise linear Bernoulli toral linked twist maps, given any <b>epsilon > 0</b> there is a Bernoulli toral linked twist map <i><b>g'</i></b> with positive Lyapunov exponents defined only on a set of measure zero such that <i><b>g'</i></b> is within <b>epsilon</b> of <i><b>g</i></b> in the <i><b>d</i></b> metric.</p
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