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Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder
Alsedà i Soler, Lluís (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Mañosas, Francesc (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Morales, Leopoldo (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Data: 2015
Resum: We extend the results and techniques from [7] to study the combinatorial dynamics (forcing) and entropy of quasiperiodically forced skewproducts 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 Sharkovski˘ı Theorem for quasiperiodically forced skew-products on the cylinder proved in [7]. Next, the notion of s-horseshoe is defined for quasiperiodically forced skewproducts 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 connectthe-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 x q − 2x q−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.
Drets: Tots els drets reservats.
Llengua: Anglès
Document: article ; recerca ; preprint
Matèria: Combinatorial dynamics ; Forcing entropy ; Irrational rotation ; Quasiperiodically forced systems on the cylinder
Publicat a: Journal of Mathematical Analysis and Applications, Vol. 429 (2015) , p. 542-561, ISSN 0022-247X

DOI: 10.1016/j.jmaa.2015.03.038


19 p, 430.4 KB

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