By P. C. Sabatier

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G ≥ 0) the mapping β → ξn (β, l) is monotonically decreasing for any ﬁxed numbers l and n, with strict monotonicity if l ≥ 0 and n ≥ 1 since g(0) = 1. e. ξn ( 23 , l) < − α1 . 1) In order to see this, note that ξ0 (β, l) is always smaller than 3, such that ξ0 (β, l) − xα ≤ 2 − √2α . 7) and g(0) = 1 we obtain ξ1 ( 32 , l) = F (ξ0 (β, l)) − 3 2 · g(0) ≤ xα + 2− √2α √ 2 α − 3 2 = √3 α − 1 α − 1 2 . 2) the right side is smaller than − α1 , and as T1 × [−3, − α1 ) is always mapped into itself this proves our claim.

Furthermore, no other invariant graph can intersect N . (i) On the one hand, there obviously exist three invariant graphs at β = 0, namely the constant lines corresponding to the three ﬁxed points. As these are not neutral, they will also persist for small values of β. On the other hand consider β = C. As we assumed that g takes the maximum value of 1 at least for one θ0 ∈ T1 , the point (θ0 , C) is mapped into M . ) But as we have seen, any point in M is attracted to ϕ− independent of β. Thus there exists an orbit which starts above the upper bounding graph and ends up converging to ϕ− .

However, as treating them with the methods presented here would even need some additional modiﬁcations, we refrain from doing so. For the case of the qpf Arnold circle map, the situation is completely diﬀerent. Here it is just not possible to apply our results. 7) is chosen, the maximal expansion rate is always at most two. Further, for any interval of ﬁxed length the uniform contraction rate also remains bounded. Although the derivative goes to zero at θ = 12 if α is close to 1, a strong contraction only takes place locally.