摘要
Let (t(n)) n >= 0 be the well-known +/- 1 Thue-Morse sequence @@@ +1,-1,-1, +1,-1, +1,+1,-1,.... @@@ Since the 1982-1983 work of Coquet and Dekking, it is known that Sigma(k<n) t(k)e(2k pi i/3) is strongly related to the famous Koch curve. As a natural generalization, for m is an element of N, we use Sigma(k<n) delta(k)e(2k pi i/m) to define the generalized Koch curve, where (delta(n)) n >= 0 is the generalized Thue-Morse sequence defined to be the unique fixed point of the morphism @@@ +1 -> +1, +delta(1),..., +delta(m), @@@ -1 -> -1, -delta(1),..., -delta(m) @@@ beginning with delta(0) = +1 and delta(1),..., delta(m) is an element of {+1,-1}, and we prove that generalized Koch curves are the attractors of the corresponding iterated function systems. For the case that m >= 2, delta(0) = ... = delta([m/4]) = +1, delta([m/4]+1) = ... = delta(m-[m/4]-1) = -1 and delta(m-[m/4]) = ... = delta(m) = +1, the open set condition holds, and then the corresponding generalized Koch curve has Hausdorff, packing and box dimension log (m + 1)/log vertical bar Sigma(m)(k=0) delta(k)e(2kpi/m)vertical bar, where taking m = 3 and then delta(0) = +1, delta(1) = delta(2) = -1, delta(3) = +1 will recover the result on the classical Koch curve.