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Suppose we have an iterated map $g(x)=x+f(x)$ where $f$ is a flat function at the origin (i.e. smooth, with all derivatives vanishing at zero). The standard stability test (for the origin) of looking at higher derivatives (note that $g'(0)=1$, so one needs to look into higher order derivatives till something non-vanishing pops up, and then depending on whether the order is even or odd and whether the derivative is positive or negative, one gets information about stability, instability or semi-stability as usual business) is non-conclusive here. Are there other methods to determine the dynamics of this flat perturbation of the identity map? Is it going to depend on the specific form of $f$? My guess is yes, and depends on whether it lowers or lifts the graph of the identity below or above the diagonal. Looking just at the right neighbourhood of $0$ for simplicity, if $f=e^{-\frac{1}{x^2}}$ we should get unstable as 0 seems repelling from a cobweb diagram point of view, if $f=-e^{-\frac{1}{x^2}}$ we should get asymptotic stability for similar reasons. This reasoning is very informal of course, but does it go in the right direction?

xyz
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  • I'm not sure you can say something much better than looking at the sign of $f$. – Albert Oct 07 '21 at 11:54
  • See https://math.stackexchange.com/questions/3215/convergence-of-sqrtnx-n-where-x-n1-sinx-n and the link bar there for an exhaustive discussion of $g(x)=\sin(x)$ which is of the prescribed form. A similar treatment is possible for all $f(x)=-ax^n+O(x^{n+1})$. If $f$ is flatter than polynomial, another trick might become available. – Lutz Lehmann Oct 07 '21 at 19:02
  • @LutzLehmann Thanks for the reference, I'll have a look. Unfortunately $\sin(x)$ is not of the prescribed form since the perturbation is not flat (by flat function I mean that derivatives of all orders vanish at a given point, the origin here). What makes the case of $\sin$ very manageable is precisely the fact that the perturbation is analytic, and so very easy to work with. On the other hand I constructed my example to stretch things in the opposite direction, where power series and Taylor estimates are not possible. – xyz Oct 07 '21 at 22:07

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