A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps (Q950602)

The authors study the one--dimensional quasi--periodically forced dynamical system \[ \overline x = f_\mu(x\,\vartheta), \quad \overline \vartheta = \vartheta + \omega, \quad\quad (x,\vartheta) \in \mathbb{R} \times \mathbb{T} \] where \(\omega\) is irrational or subject to a Diophantine condition with respect to the circle \(\mathbb{T}\). After preparatory results concerning reducibility, normal forms, Lyapunov exponents, spectrum of the so called transfer operators the authors prove theorems about fractalization of invariant solution curves. Fractalization of a solution curve is defined by blowing up derivatives while the function itself remains bounded. The final section contains some numerical experiments showing the relevance of the results.