Analytic relations on a dynamical orbit [article]

Thomas Scanlon
2008 arXiv   pre-print
Let (K,|·|) be a complete discretely valued field and f: B_1(K,1) → B_1(K,1) a nonconstant analytic map from the unit back to itself. We assume that 0 is an attracting fixed point of f. Let a ∈ K with _n →∞ f^n(a) = 0 and consider the orbit O_f(a) := {f^n(a) : n ∈ N}. We show that if 0 is a superattracting fixed point, then every irreducible analytic subvariety of B_n(K,1) meeting O_f(a)^n in an analytically Zariski dense set is defined by equations of the form x_i = b and x_j = f^ℓ(x_k). When
more » ... is an attracting, non-superattracting point, we show that all analytic relations come from algebraic tori.
arXiv:0807.4162v1 fatcat:r5lczq2u3nbi5d5cjsjjmhnppy