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I hope this question is well-posed.

Let (X, f) be a discrete dynamical system such that every x in X has finite period, i.e. there is some n such that f^n(x) = x. Let Div(X) be the free abelian group on the orbits of X. When X is a nonsingular algebraic curve over the algebraic closure of a finite field k and f is the Frobenius map, Div(X) is naturally isomorphic to the group of fractional ideals of k(X) (at least, I think; correct me if I'm wrong). There is a distinguished subgroup Prin(X) consisting of the preimage of the principal ideals, and Div(X)/Prin(X) is the divisor class group.

Is there a canonical definition of Prin(X) for general dynamical systems? If not, how much extra structure does X need to have for a construction like this to make sense and give some kind of useful information about X?

The case I'm interested in is that X is the set of aperiodic closed walks on a finite graph with a distinguished point and f moves the distinguished point.

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It seems like a problem for a general discrete dynamical system would be the identicalness of the points. In the case of an algebraic curve over the algebraic closure of a finite field, the points are endowed with extra structure from the notion of functions on the set, which is exactly where Prin(X) comes from. At least in your case, the discrete set still has a notion of 'closeness' that I guess might lend enable something like Prin(X), but I don't see any candidate. – Greg Muller Oct 21 '09 at 5:40

This isn't pretending to be an answer to your question, but I believe that in the context of certain chip-firing games on graphs you can push the analogy with algebraic curves pretty far (although for various reasons which I don't understand the correct field seems to be C rather than a field of finite characteristic). In particular you can define a notion of principal divisors, although I don't know whether it's exactly what you want.

See for instance Postnikov and Shapiro, or more explicitly Baker and Norine and obviously the cited papers of both.

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My understanding is that those connections go through the Laplacian, which I'm not interested in. – Qiaochu Yuan Oct 21 '09 at 15:28

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