Abstract
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 d metric.
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