Rotation numbers and moduli of elliptic curves

Complex rotation numbers

“…The case of hyperbolic diffeomorphisms was dealt first by Yu. Ilyashenko and V. Moldavskis [5], then this result was improved by N.Goncharuk [4]. For exact statements of these results, see Section 2.…”

“…He also studied the behavior of τ f and obtained some formulas and estimates on its derivatives; in particular, he proved that τ f is injective on Ω s provided that f is close to rotation. The case of parabolic cycles was studied by J.Lacroix (unpublished) and N.Goncharuk [4] independently. The case of hyperbolic diffeomorphisms was dealt first by Yu.…”

Self-similarity of bubbles

“…Two independent proofs of this conjecture were given in [11] and [10]. This statement does not hold if f + ω 0 is hyperbolic, as was proved in [6]; this result was strengthened in [5]. The case of a diffeomorphism with parabolic cycles was studied by J.Lacroix (unpublished) and in [5].…”

“…The analyticity ofτ f in the last subcase below is not included in [2,Main theorem], but it constitutes [5,Theorem 1.2], see also [2,Theorem 2].…”

Complex rotation numbers: bubbles and their intersections

“…The rest of the construction is analogous to the case of rot f = 0. Theorem 4 ( [5]; see also [4,Sec. 6]).…”