Scaling properties in dynamics of non-analytic complex maps near the accumulation point of the period-tripling cascade

The accumulation point of the period-tripling bifurcation cascade in complex quadratic map was discovered by Golberg, Sinai, and Khanin (Russ.Math.Surv. 38:1, 1983, 187), and independently by Cvitanović and Myrheim (Phys.Lett. A94:8, 1983, 329). As we argue, in the extended parameter space of smooth maps, not necessary satisfying the Cauchy – Riemann equations, the scaling properties associated with the period-tripling are governed by two relevant universal complex constants. The first one is δ1 ∼= 4.6002 − 8.9812i (in accordance with the mentioned works), while the other one, responsible for the violation of the analyticity, is found to be δ2 ∼= 2.5872 + 1.8067i. It means that in the extended parameter space the critical behaviour associated with the period-tripling cascade is a phenomenon of codimension four. Scaling properties of the parameter space are illustrated by diagrams in special local coordinates. We emphasize a necessity for a nonlinear parameter change to observe the parameter space scaling. PACS numbers: 05.45.-a, 05.10.Cc, 05.45.Df.