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Recently, I read a paper about discrete Schrödinger operator. There is a map related to trace map from $C^3$ to $C^3$ as follows:

$$T(x,y,z)=(y,z,yz-x).$$

We can calculated that $T$ has the folliwng invariant surface $$x^2+y^2+z^2-xyz-2=D.$$

Of course in this case, we have some geometry explanations for it.

My question is that give a general diffeomorphism $T$ from $C^n$ to $C^n$, is it possible to given some criterion get invariant surface or get rid off invariant surface?

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  • $\begingroup$ What sort of hypothesis are you willing to work with? By the Poincare--Dulac theorem, there is a stable condition on biholomorphisms that fix a point under which they also fix a hypersurface: no resonant polynomials of eigenvalues and eigenvalues lying in the Poincare domain. $\endgroup$
    – Ben McKay
    Dec 5, 2014 at 11:26

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