Coding Rule for Periodic Orbits in the One-dimensional Map

抄録

A new coding rule for periodic orbits in unimodal one-dimensional maps is derived. The best-known example of a family of unimodal maps is the logistic map. The band merging is observed in the bifurcation diagram of the logistic map. Let a_m^k(k ≥ 1) be the critical value at which 2^k-band merges into 2^{k - 1}-band. At a > a_m⁰, the diverging orbit appears and thus 1-band disappears. The relations a_m^{k + 1} < a_m^k for k ≥ 0 hold. Let s_q be the code for periodic orbit of period q in the parameter interval (a_m¹, a_m⁰]. Assume that the code s_q represented by symbols 0 and 1 is known. In the interval (a_m^{k + 1}, a_m^k], there exists the periodic orbit of period 2^k × q (k ≥ 1). Let its code be s_2^k×q. Let 𝒟 be the doubling operator defined by the substitution rules as 0 ⇒ 11 and 1 ⇒ 01. The following coding rule is derived. Operating k times of 𝒟 to s_q, the code s_2^k×q is determined.