Figure 3.

(a) Iteration time dependence of positions θ_t in logarithmic scales for the orbit with initial condition θ₀ = 0 at ${\mathrm{\Omega }}_{\infty }^{\prime }$ of the critical circle map K = 1. The labels indicate iteration time t, the blue line goes through positions at times of the form $F_{2n}$, while the dotted and dashed lines do similarly at times of the form $2F_{2n}$ and $F_{2n}+F_{2n-2}$, respectively. The arrows indicate the distances equivalent to the principal diameters $d_{2n,0}$. (b) The positions θ_t of the trajectory in (a) at selected iteration times $t=F_{2n}-1$ (black dots) reproduced by Eq. (17) with α = 1.948997 (red). The inset shows the same data plotted in ${\ln }_{{\alpha }^{\prime }}$ scale, p_k = 1/θ_k, p_min = 1 and α^′ = 0.051003. The straight line evidences the (time) extensivity of the entropy in Eq. (13). See text for description.