Some Properties of Real and Complex Brjuno Functions

Quasi-periodic motions in strongly dissipative forced systems

“….+|ν d | and Z d * := Z d \{0}, and for some positive constants γ 0 and τ , then it also satisfies (1.3). Recently, the Bryuno condition has received a lot of attention in the theory of small divisor problems; see for instance [13,10,11,12,15] and papers cited therein.…”

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“…This series diverges on a set of measure 0. Outside it converges to a measurable function, continuous at irrational points, with period 1 [Marmi et al 2006].…”

Remarks on modular symbols for Maass wave forms

“…According to [Lewis and Zagier 2001, p. 255], the function This series diverges on a set of measure 0. Outside it converges to a measurable function, continuous at irrational points, with period 1 [Marmi et al 2006].…”

“…In the last subsections I show that the respective transfer operator can be treated as the "Hecke operator T 1 " corresponding to the "prime" in characteristic zero, and that the modified Brjuno function in the theory of linearization of holomorphic maps [Marmi et al 2001;2006] can be used for the calculation of derivatives of classical Dirichlet series related to cusp forms, replacing the eta function appearing in [Goldfeld 1995].…”

“…In fact, an interesting theory is developed/reviewed in [Marmi et al 2001;2006] for another Brjuno function B(ξ ), which differs from b(ξ ) by O(1), but satisfies a functional equation and has a complex version closely resembling some constructions in the theory of modular forms. In our context, it can be used for calculation of the derivative of some classical L-series at certain points.…”