it's an interaction between dynamics and algebra.(adsbygoogle = window.adsbygoogle || []).push({});

let f and g be elements of C[x] where C is complex numbers (or any alg. closed field).

denote the nth iterate of f by f^n, though that notation is often also used for the nth power of f. (this is where the dynamics of f is involved.)

consider the ideal (and affine variety) generated by f-g, f^2-g, f^3-g, ... . denote this by I.

the hilbert basis theorem states that this can be generated by a finite number of elements in I. denote this finite generating set by [f,g], not to imply that the generators are f and g. then the variety associated with I is the same as the one associated with <[f,g]>, the "span" of [f,g]. denote the variety by V[f,g].

i want to investigate properties of V[f,g] depending on f and g. the easiest case is if f is a constant function. then its hilbert dimension is easy to calculate. if f is linear, things get a lot more interesting already. i imaging that for most or all g, V[f,g] is empty, but i'm not sure.

any thoughts?

well, i was hoping this approach might shed some light on the dynamics of f and my ultimate goal would be to say something like f "converges" to g if V[f,g] is of maximal dimension and "does not converge" if for all g, V[f,g] is empty. something like that.

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# Dynamics and algebra

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