Bounds for maps of an interval with one critical point of inflection type. II (Q1579079)

[Part I, cf. the first author, Fundam. Math. 157, 287-298 (1998; Zbl 0915.58028).] Recently quite a few papers appeared proving complex bounds, local connectivity of Julia sets, absence of invariant line fields and quasisymmetric rigidity for real polynomial maps. This paper deals with a class, which contains smooth covering maps of the circle with a unique critical point of inflection type. In particular, this class includes (generalized Arnold) maps of the form \(f(x)=kx +a+b\sin 2\pi x\pmod 1\), where \(k\in N\), \(k\geq 2\) and \(B\) is chosen in such a way, that \(f\) possesses a cubic critical point. Here the authors study the metric property for maps with points of inflection. Note that to prove real and complex bounds for maps with a critical point of inflection type is more involved than for maps with a folding critical point.