Fat Fractals on the Energy Surface

For a closed system of two coupled nonlinear oscillators, chaotic orbits are punctuated by holes associated with stable periodic orbits. For the corresponding class of Hamiltonian maps we demonstrate that the combined area for all holes of size $\epsilon$ or greater scales as a power law with exponent $\beta$ and asymptotic area $0<\mathrm{}\mu <1\mathrm{}$. In contrast to previous results, this is a global scaling property, valid for a set of positive Lebesgue measure. It suggests that these chaotic orbits are fat fractals, i.e., Cantor-set-like objects of positive area. We numerically compute lower bounds on their area.