Invariant curves for exact symplectic twist maps of the cylinder with Bryuno rotation numbers

Abstract

Since Moser's seminal work it is well known that the invariant curves of smooth nearly integrable twist maps of the cylinder with Diophantine rotation number are preserved under perturbation. In this paper we show that, in the analytic class, the result extends to Bryuno rotation numbers. First, we will show that the series expansion for the invariant curves in powers of the perturbation parameter can be formally defined, then we shall prove that the series converges absolutely in a neighbourhood of the origin. This will be achieved using multiscale analysis and renormalisation group techniques to express the coefficients of the series as sums of values which are represented graphically as tree diagrams and then exploit cancellations between terms contributing to the same perturbation order. As a byproduct we shall see that, when perturbing linear maps, the series expansion for an analytic invariant curve converges for all perturbations if and only if the corresponding rotation number satisfies the Bryuno condition.

Stable dynamics in forced systems with sufficiently high/low forcing frequency

“…Note that, in order to prove just boundedness of the solutions, it would be enough to prove the existence of an invariant circle of constant type (as sometimes done in the literature, see for instance [11]; see also the comments in [17]). On the contrary, to prove that the persisting tori have large measure for ε small, a milder Diophantine condition is required; one could even allow a Bryuno condition on the rotation number [8], as done in [16], but this would not increase appreciably the measure of the invariant curves.…”

Regularity properties of $k$-Brjuno and Wilton functions

“…For instance it is conjectured that the Brjuno condition is optimal for the existence of real analytic invariant circles in the standard family [Mac88,Mac89,MS92]. See also [BG01,Gen15] and references therein for related results.…”

Some remarks on the optimality of the Bruno-Rüssmann condition

“…To describe his result, let us first remark that α ∈ B obviously implies that α ∈ R in the sense that lim n→+∞ log q n+1 q n = 0 (R) but clearly the converse is not true. The condition that α ∈ R is in fact the necessary and sufficient condition for the linearized problem (the so-called cohomological equation) to have a solution in the analytic topology ( [Rüs75] [Gen15] and other references by the same author) it is stated that α ∈ B is optimal for the existence of an analytic invariant circle for the standard map in the perturbative regime which depends analytically in the small parameter; we would like to point out that this statement is incorrectly deduced from results of Marmi ([Mar90]) and Davie ([Dav94]) and thus the optimality of α ∈ B for the standard map is still an open question (see [MM00] where this observation is also made). number α.…”

“…The condition that α ∈ R is in fact the necessary and sufficient condition for the linearized problem (the so-called cohomological equation) to have a solution in the analytic topology ([Rüs75]). Using results of Mather ([Mat86], [Mat88]) and Herman ([Her83]), Forni proved that if an integrable twist map has an invariant curve with rotation number α / ∈ R, then there exists arbitrarily small analytic perturbation for which there are no (necessarily Lipschitz) invariant curves with rotation 1 Unfortunately, at several places in the literature (for instance [Gen15] and other references by the same author) it is stated that α ∈ B is optimal for the existence of an analytic invariant circle for the standard map in the perturbative regime which depends analytically in the small parameter; we would like to point out that this statement is incorrectly deduced from results of Marmi ([Mar90]) and Davie ([Dav94]) and thus the optimality of α ∈ B for the standard map is still an open question (see [MM00] where this observation is also made). number α.…”