On a result by Carvalho concerning S̆arkovskii's order

AbstractIt is well known that the celebrated S̆arkovskii's Theorem [4] (cf. also [1]) defines a total ordering ▷ on the set N∗ of positive integers (the S̆arkovskii's order) such that, if f is a continuous map of an interval J of the real line R into itself with a periodic orbit of period p, and p ▷ q, then f has a periodic orbit of period q. The S̆arkovskii's order has a minimum, namely the period 3. Recently, Carvalho [2] has observed that the orbits considered in S̆arkovskii's Theorem are of a special kind with respect to the natural order on R of their points. Thus, in [2], he extended the classical S̆arkovskii's order below the standard minimum joining a sequence of so-called “n-step orbits”: an n-step orbit of a continuous map f: J → J (where J is an interval not reduced to a single point of the real line) is a periodic orbit (x1, x2, …, xn) of f with period n ⩾ 1, consisting of n distinct points x1 < x2 < ··· < xn with xj + 1 = f(xj) for j = 1, …, n − 1, and x1 = f(xn)