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Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder
Alsedà, 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.
Nota: Número d'acord de subvenció MINECO/MTM2008-01486
Nota: Número d'acord de subvenció MINECO/MTM2011-26995-C02-0
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


Preprint
19 p, 430.4 KB

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