We study circle homeomorphisms f ∈ C 2 (S 1 {x b }) whose rotation number ρ f is irrational, with a single break point x b at which f ′ has a jump discontinuity. We prove that the behavior of the ratios of the lengths of any two adjacent intervals of the dynamical partition depends on the size of break and on the continued fraction decomposition of ρ f . We also prove a result analogous to Yoccoz’s lemma on the asymptotic behaviour of the lengths of the intervals of trajectories of the renormalization transformation R n (f).

Some remarks on the geometry of circle maps with a break point / Simsek, Necip; Dzhalilov, Akhtam; Musso, Emilio. - In: FILOMAT. - ISSN 0354-5180. - ELETTRONICO. - 32:16(2018), pp. 5549-5563. [10.2298/FIL1816549S]

Some remarks on the geometry of circle maps with a break point

Musso, Emilio
2018

Abstract

We study circle homeomorphisms f ∈ C 2 (S 1 {x b }) whose rotation number ρ f is irrational, with a single break point x b at which f ′ has a jump discontinuity. We prove that the behavior of the ratios of the lengths of any two adjacent intervals of the dynamical partition depends on the size of break and on the continued fraction decomposition of ρ f . We also prove a result analogous to Yoccoz’s lemma on the asymptotic behaviour of the lengths of the intervals of trajectories of the renormalization transformation R n (f).
2018
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2727574
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