A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps

We introduce a scenario for the fractalization of invariant curves for a special class of quasi-periodically forced 1-D maps. In this situation, a smooth invariant curve becomes increasingly wrinkled when its Lyapunov exponent goes to zero, but it keeps being smooth as long as its exponent is negative. It is remarkable that the curve becomes so wrinkled that numerical simulations may not distinguish the curve from a strange attracting set.
    Moreover, we show that a nonreducible invariant curve with a positive Lyapunov exponent is not persistent in a general quasi-periodically forced 1-D map. We also derive some new results on the behaviour of the Lyapunov exponent of an invariant curve w.r.t. parameters.
The paper contains some numerical examples. One of them is based on the quasi-periodically forced logistic map, where we show numerically that the fractalization of an invariant curve of this system may fit into our scenario.