On an extension of Sarkovskii's order (Q1118706)

Let f:J\(\to J\) be continuous, J an interval in R. For \(x\in J\) the orbit \(\theta_ f(x)=\{x,f(x),f^ 2(x),...\}\) is n-periodic if n is the least integer with \(f^ n(x)=x.\) There are (n-1)! types of n-periodic orbit depending solely on the relative position of points \(x,f(x),...,f^{n- 1}(x)\) with respect to the natural ordering. An n-periodic orbit is \(n_ 1\)-periodic if just one point is below the diagonal (x,x). The following extension of Sharkovskij's theorem is proved: Consider the set \(N'=\{1,2,3,4,5,6,...,3_ 1,4_ 1,5_ 1...\}\) with the ordering \[ 1\triangleleft 2\triangleleft 4\triangleleft 8\triangleleft \quad...\quad \triangleleft 7.2^ j\triangleleft 5.2^ j\triangleleft 3.2^ j\triangleleft \quad... \] \[ ...\quad \triangleleft 7.2^{j- 1}\triangleleft 5.2^{j-1}\triangleleft 3.2^{j-1}\triangleleft \quad...\quad \triangleleft 7\triangleleft 5\triangleleft 3_ 1\triangleleft 4_ 1\triangleleft 5_ 1\triangleleft \quad...\quad, \] where \(j=1,2,3,..\).. If \(p,q\in N'\) and \(p\triangleleft q\), then the existence of a q-orbit implies the existence of a p-orbit for any continuous map of the interval.