A Polynomial Bound for the Lap Number

In this note, we show a polynomial bound for the growth of the lap number of a piecewise monotone and piecewise continuous interval map with nitely many periodic points. We use Milnor and Thurston's kneading theory with the coordinates of Baladi and Ruelle, which are useful for extending the theory to the non continuous case. We say that f : 0; 1] ! 0; 1] is piecewise strictly monotone piecewise continuous (PSMC) if there is a nite number of so-called critical points c 1 < < c r 2 (0; 1) such that f is continuous and strictly monotone on (c i ; c i+1) and on (0; c 1) and (c r ; 1). The composition of two PSMC maps is also PSMC, usually with a diierent set of critical points. We denote by C f = fc i ji = 1 rg the set of critical points of f (we choose C f to be minimal in the obvious sense) and by p f (n) the minimal number of intervals in which f n (= f f n times) is continuous and strictly monotone. We call p f (n) the lap number of f n. Clearly, p f (n) = #C f n + 1. We say that x 2 0; 1] is periodic with period m = m(x) if f m (x) = x. It is repelling if there is > 0 such that for jx ? yj < and 0 m(x), lim n!1 f ?(n+`) y = f ` x, where f ?1 denotes the local inverse branch of f. Set : 0; 1] ! f0; 1g by (x) = 0 if x 2 C f , and (x) = 1 depending on whether f is increasing or decreasing at x = 2 C f. Let = 0 if ((0; c 1)) ((c r ; 1)) = ?1, and 1 otherwise. In this note, we prove the following proposition: Theorem 1. Let f : 0; 1] ! 0; 1] be a PSMC map, with a nite number of periodic points. Then, p f (n) Cst n s , where s = r + ` + + 1, r = #C f ; ` = #frepelling periodic orbits of fg, and Cst is a constant depending only on s and the number of periodic points of f. The error term is probably an artifact of the proof …