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

Abstract

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 $\log(s).$ 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 $\tau.$ This implies that if the period of $\tau$ is $2^n q$ with $n \ge 0$ and $q \ge 1$ odd, then $h(F) \ge \tfrac{\log(\lambda_q)}{2^n}$, 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.