Rotation number of contracted rotations

Abstract : Let 0 < lambda< 1. We consider the one-parameter family of circle lambda-affine contractions f(delta): x is an element of [0,1] -> lambda x + delta mod 1, where 0 <= delta < 1. Let rho be the rotation number of the map f(delta). We will give some numerical relations between the values of lambda,delta and rho, essentially using Hecke-Mahler series and a tree structure. When both parameters lambda and delta are algebraic numbers, we show that rho is a rational number. Moreover, in the case lambda and delta are rational, we give an explicit upper bound for the height of rho under some assumptions on lambda.
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Submitted on : Friday, March 1, 2019 - 9:19:26 AM
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Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of modern dynamics, American Institute of Mathematical Sciences, 2018, 12 (1), pp.175-191. ⟨10.3934/jmd.2018007⟩. ⟨hal-02053125⟩



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