Coding Rule for Periodic Orbits in the One-dimensional Map

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 ak m (k ≥ 1) be the critical value at which 2k-band merges into 2k−1-band. At a > a0 m , the diverging orbit appears and thus 1-band disappears. The relations ak+1 m < a k m for k ≥ 0 hold. Let sq be the code for periodic orbit of period q in the parameter interval (a1 m, a 0 m]. Assume that the code sq represented by symbols 0 and 1 is known. In the interval (ak+1 m , a k m], there exists the periodic orbit of period 2 k × q (k ≥ 1). Let its code be s2k×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 sq , the code s2k×q is determined.