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

We extend the results and techniques from [7] to study the combinatorial dynamics (forcing) and entropy of quasiperiodically forced skew-products on the cylinder. For these maps we prove that a cyclic permutation τ forces a cyclic permutation ν as interval patterns if and only if τ forces ν as cylinder patterns. This result gives as a corollary the Sharkovskiĭ Theorem for quasiperiodically forced skew-products on the cylinder proved in [7]. 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 τ, then h(F)≥h(fτ), where fτ denotes the connect-the-dots interval map over a periodic orbit with pattern τ. This implies that if the period of τ is 2nq with n≥0 and q≥1 odd, then h(F)≥log⁡(λq)2n, where λ1=1 and, for each q≥3, λq is the largest root of the polynomial xq−2xq−2−1. Moreover, for every m=2nq with n≥0 and q≥1 odd, there exists a quasiperiodically forced skew-product on the cylinder Fm with a periodic orbit of period m such that h(Fm)=log⁡(λq)2n. This extends the analogous result for interval maps to quasiperiodically forced skew-products on the cylinder.