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

Abstract

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 Sharkovskiȋ 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(f τ ), where f τ denotes the connect-the-dots interval map over a periodic orbit with pattern τ . This implies that if the period of τ is 2 n q with n ≥ 0 and q ≥ 1 odd, then h(F ) ≥ log(λq ) 2 n , where λ 1 = 1 and, for each q ≥ 3, λ q is the largest root of the polynomial x q − 2x q−2 − 1. Moreover, for every m = 2 n q with n ≥ 0 and q ≥ 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 ) = log(λq ) 2 n . This extends the analogous result for interval maps to quasiperiodically forced skew-products on the cylinder.