A polynomial bound for the lap number (Q1431472)

The goal of this note is proving the following theorem: let \(f:[0,1]\to [0,1]\) be a piecewise strictly monotone piecewise continuous map with \(q\) periodic points, \(q< \infty\). Then the lap number of \(f^n\), that is \(P_f(n)\leq C(s,q)\cdot n^s\), where \(s= r+\ell+\nu+1\), where \(r=\# C_f\), \(\ell=\#\) (repelling periodic orbits of \(f\)) and \(C(s,q)\) is a constant depending only on \(s\) and \(q\). Here, \(\nu= 0\) if \(\varepsilon((0,c_1))\cdot\varepsilon((c_r,1))= -1\) and \(1\) otherwise, with \(\varepsilon:[0,1]\to \{0,\pm 1\}\) by \(\varepsilon(x)= 0\) if \(x\in C_f\), and \(\varepsilon(x)= \pm1\) depending on whether \(f\) is increasing or decreasing at \(x\not\in C_f\) (by \(C_f\) is denoted the set of the critical points of \(f\)).