GENERALIZED KOCH CURVES AND THUE–MORSE SEQUENCES.

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 ∑ k < n t k e 2 k π i 3 is strongly related to the famous Koch curve. As a natural generalization, for m ∈ ℕ , we use ∑ k < n δ k e 2 k π i m to define the generalized Koch curve, where (δ n) n ≥ 0 is the generalized Thue–Morse sequence defined to be the unique fixed point of the morphism + 1 ↦ + 1 , + δ 1 , ... , + δ m , − 1 ↦ − 1 , − δ 1 , ... , − δ m beginning with δ 0 = + 1 and δ 1 , ... , δ m ∈ { + 1 , − 1 } , and we prove that generalized Koch curves are the attractors of the corresponding iterated function systems. For the case that m ≥ 2 , δ 0 = ⋯ = δ ⌊ m 4 ⌋ = + 1 , δ ⌊ m 4 ⌋ + 1 = ⋯ = δ m − ⌊ m 4 ⌋ − 1 = − 1 and δ m − ⌊ m 4 ⌋ = ⋯ = δ m = + 1 , the open set condition holds, and then the corresponding generalized Koch curve has Hausdorff, packing and box dimension log (m + 1) / log | ∑ k = 0 m δ k e 2 k π i m | , where taking m = 3 and then δ 0 = + 1 , δ 1 = δ 2 = − 1 , δ 3 = + 1 will recover the result on the classical Koch curve. [ABSTRACT FROM AUTHOR]