Volume 153, number 6,7 PHYSICS LETTERS A 11 March 1991 The ordering of critical periodic points in coordinate space by symbolic dynamics Shou-Li Peng 1 and Li-Shi Luo 2 School ofPhysics, Georgia Institute of Technology, Atlanta, GA 30332-0430, USA Received 25 June 1990; revised manuscript received 4 January 1991; accepted for publication 7 January 1991 Communicated by D.D. HoIm In this paper, we study the symbolic dynamics equipped with the DGP product “*“ and the shift operator ~ conjugate to the unimodal mapping f~:IF-.I over the interval I = [— 1, 1]. We prove two theorems, and give an explicit formula for the rule of ordering the points on the n-tupling periodic trajectory {ço’~(i”~(p+ I)) } in coordinate space with arbitrarily large p and n. Until recently, to calculate the fractal dimension of critical periodic points of unimodal maps with significant large period has been a difficult task, because the computing time for the ordering of the points on the n-tupling periodic trajectory increases exponentially with n. Such a calculation is justified because it is believed that there may exist a new global regularity of fractal dimensions on critical points [1]. Moreover, the thermodynamical formalism of dimension of complicated objects has received much attention [2—71.In this formalism, one of the important and essential aspects is how to calculate the partition function of a complex system by using the ordering of points on trajectories in the coordinate space. Even though this can be done numerically for tra- jectories with relatively short period, their analytical ordering presents serious difficulties as shown in ref. [1]. First of all, the sequence of coordinate points for an arbitrary trajectory is an infinite one in general. Second, each coordinate point on a trajectory is represented by an infinite sequence of “symbols” in general, as ex- plained later. In this work, we obtain an analytic method to reveal the rule of the ordering of the points on arbitrary n- tupling periodic trajectories of unimodal mapping in the coordinate space. Our work in this paper is an ex- tension of a work of Feigenbaum [2] where the rule of ordering points on the period-doubling trajectories, i.e., trajectories with period 2”, n = 0, 1, 2, ..., has been derived. We extend the result to trajectories with period k” for arbitrary k= 2, 3, ..., and n =1, 2, ..., and provide rigorous proofs. Our approach in this work involves symbolic dynamics and some elementary results in number theory. Consider a unimodal mapping [81 f, 4:Ii-~I over the interval I = [— 1, 1] depending on a parameter ~i. The location of the unique maximum of f~ can be normalized to be the origin. For an arbitrary point x0eI, the set O,.(xo)=f~(xo)}~{x~} is called a trajectory with initial point x0. The symbolic dynamics can be easily in- troduced by discriminating among the three possible cases x~ <0, x,, = 0 and x~>’0. Therefore, each trajectory can be associated with an infinite sequence of three symbols, L, C and R: a=a1a2...a1... , (1) where ac {L, C, R} is determined by the following rule: Permanent address: Department of Physics, Yunnan University, Kunming, Yunnan Province, China. 2 Center for Nonlinear Studies, MS-B258, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 0375-9601/91/S 03.50 © 1991 — Elsevier Science Publishers By. (North-Holland) 345