Do Arnold tongues really constitute a fractal set?

We review and analyze the main features of fractal sets, as well as argue that the geometric property called fractality is not a well defined concept in the literature. As an example, we consider the mode locking phenomenon exhibited by the Sine Circle Map, concerning those sets in parameter space called Arnold Tongues and Devil's Staircase (more exactly, their complements). It is shown that well-known results for the fat fractal exponent of the ergodic region turn out to be valid only in a tiny region of parameter space where ρ ∈ [0,a], a → 0, being ρ the winding number. A careful geometric analysis shows that a misleading simplification has led to a loss of information that hindered relevant conclusions. We propose an alternative, broader and more rigorous approach in that it selects a generic interval from the whole domain which excludes the mentioned restricted region. Our results reveal that the measure shows a different dependence on tongue widths, so we argue that such a discrepancy is only possible if we assume that the set in question does not satisfy a scale-free property. Since there is no invariant exponent for the complementary set of Arnold Tongues (and consequently for that of Devil's Staircase), we conclude that the exponent found by Ecke et al.[4] does not characterize those sets as true fractals. We also consider those sets as an example that statistical criteria are insufficient to characterize any set as being a fractal; they should always follow an analysis of the topological process responsible for generating the set.