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Let P be a polynomial functor from Top to Top, by which I mean a functor of the form P(X) = ∐i ≥ 0 Si × Xi where the Si are finite sets, all but finitely many of which are empty. Corresponding to P there is an ordinary polynomial p(x) = Σi ≥ 0 |Si| xi. I believe that there is a final P-coalgebra XP in Top; the structure map XP -> P(XP) is a homeomorphism; the underlying set of XP is the final P-coalgebra in Set; and a basis for the open sets of XP is given by the preimages of the points of the targets of the maps XP -> P(n)(XP) -> P(n)(•) as n ranges over all nonnegative integers.

Examples are the Cantor space, corresponding to p(x) = 2x, and the space of binary trees, corresponding to p(x) = x^2 + 1.

My question is: if P and Q are two polynomial functors, such that neither p(x) nor q(x) is of the form x + k, and the spaces XP and XQ are homeomorphic, does it follow that the polynomials p(x) - x and q(x) - x have a common root?

More generally, consider all the spaces that can be formed starting with the collection of spaces XP (p(x) not of the form x + k) by taking disjoint unions and products. Can I assign an algebraic number to each of these spaces in a way which is a homeomorphism invariant and commutes with sums and products?

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I don't know, but one could begin to investigate your first question by considering polynomials of the form ax + k, where a is a natural number greater than 1. Maybe you already know that the answer to the question is "yes" in that case? (If so, I'd be interested to see your answer.)

A point related to this, and to the general idea about algebraic numbers, is discussed here:

There are also close relations with the theory described in arXiv:math.DS/0411343 (an improved version of which appears in Real and Complex Singularities, ed. Paunescu et al, World Scientific 2007). But there are differences too, e.g. you allow products but I didn't.

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