Algebraic Entropy of Endomorphisms Over Local One-dimensional Domains

Let R be a local one-dimensional integral domain, with maximal ideal M and field of fractions Q. Here, a local ring is not necessarily Noetherian. We consider the algebraic entropy ent(g), defined using the invariant gen, where, for M a finitely generated R-module, gen(M) is its minimal number of generators. We relate some natural properties of R with the algebraic entropies ent(g)(phi) of the elements phi is an element of Q, regarded as endomorphisms in End(R)(Q). Specifically, let R be dominated by an Archimedean valuation domain V, with maximal ideal P. We examine the uniqueness of V, the transcendency of the residue field extension V/P over R/M, and the condition for R to be a pseudovaluation domain. We get mutual information between these properties and the behavior of ent(g), focusing on the conditions ent(g)(phi) = 0 for every phi is an element of Q, ent(g)(psi) = infinity for some psi is an element of Q, and ent(g)(phi) < infinity for every phi is an element of Q.