Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder

We extend the results and techniques from \cite{FJJK} to study the combinatorial dynamics (\emph{forcing}) and entropy of quasiperiodically forced skew-products on the cylinder. For these maps we prove that a cyclic permutation $\tau$ forces a cyclic permutation $\nu$ as interval patterns if and only if $\tau$ forces $\nu$ as cylinder patterns. This result gives as a corollary the Sharkovski\u{\i} Theorem for quasiperiodically forced skew-products on the cylinder proved in \cite{FJJK}. Next, the notion of $s$-horseshoe is defined for quasiperiodically forced skew-products on the cylinder and it is proved, as in the interval case, that if a quasiperiodically forced skew-product on the cylinder has an $s$-horseshoe then its topological entropy is larger than or equals to $\log (s)\mathrm{.}$ Finally, if a quasiperiodically forced skew-product on the cylinder has a periodic orbit with pattern $\tau ,$ then $h(F)\ge h(f_{\tau }),$ where $f_{\tau }$ denotes the \emph{connect-the-dots} interval map over a periodic orbit with pattern $\tau \mathrm{.}$ This implies that if the period of $\tau$ is $2^{n}q$ with $n\ge 0$ and $q\ge 1$ odd, then $h(F)\ge \frac{\log ({\lambda }_q)}{2^n}$, where ${\lambda }_1=1$ and, for each $q\ge 3,$ ${\lambda }_q$ is the largest root of the polynomial $x^q-2x^{q-2}-1.$ Moreover, for every $m=2^{n}q$ with $n \ge 0$and$q \ge 1$ odd, there exists a quasiperiodically forced skew-product on the cylinder $F_m$ with a periodic orbit of period $m$ such that $h(F_m) = \tfrac{\log(\lambda_q)}{2^n}.$ This extends the analogous result for interval maps to quasiperiodically forced skew-products on the cylinder.