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

TL;DR

This paper extends the understanding of forcing, entropy, and periodic patterns in quasiperiodically forced skew-products on the cylinder, establishing new results that connect interval dynamics with cylinder dynamics.

Contribution

It proves the equivalence of forcing relations between cyclic permutations for interval and cylinder patterns, and extends entropy and periodic orbit results from interval maps to cylinder skew-products.

Findings

Forcing relations for permutations are equivalent between interval and cylinder maps.

Presence of an s-horseshoe implies entropy at least log(s).

Constructed examples show entropy bounds are sharp for given periods.

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.