GENERALIZED KOCH CURVES AND THUE–MORSE SEQUENCES

Abstract

Let [Formula: see text] be the well-known [Formula: see text] Thue–Morse sequence [Formula: see text] Since the 1982–1983 work of Coquet and Dekking, it is known that [Formula: see text] is strongly related to the famous Koch curve. As a natural generalization, for [Formula: see text], we use [Formula: see text] to define the generalized Koch curve, where [Formula: see text] is the generalized Thue–Morse sequence defined to be the unique fixed point of the morphism [Formula: see text] [Formula: see text] beginning with [Formula: see text] and [Formula: see text], and we prove that generalized Koch curves are the attractors of the corresponding iterated function systems. For the case that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the open set condition holds, and then the corresponding generalized Koch curve has Hausdorff, packing and box dimension [Formula: see text], where taking [Formula: see text] and then [Formula: see text] will recover the result on the classical Koch curve.