A Family of 2א1 Logarithmic Functions of Distinct Growth Rates

Let G be an ordered abelian group and R((G)) the field of generalized series. In [KS05] we define exponential functions on the κ-maximal subfield R((G))κ; the subfield of series with support of cardinality bounded by a regular uncountable cardinal κ. The purpose of this note is to give a natural functional interpretation of the formal construction of [KS05]. To this end, the group G considered here is determined by a totally ordered set Γ of (germs at +∞ of) real valued functions. More precisely, we construct such a Γ of cardinality א1. We show that Γ is isomorphic to the lexicographic product א1×Z×Z; which admits 2א1 automorphisms of pairwise distinct orbital growth. We associate to each such automorphism a well defined logarithmic function on the field R((G))א1 , where R((G))א1 is the field of generalized series with countable support, real coefficients and monomials in the group G defined by Γ. We show that distinct automorphisms induce logarithmic functions of distinct growth rates. This result is particularly interesting in connection to the extensive investigations of the asymptotic scale undertaken in the early 20th Century, see [H1909a], [Har1954] and [Koj09]. In contrast to the notation of [KS05] and [Kuh00], G here is written in multiplicative notation (which is better adapted to the functional setting). For the convenience of the reader, we summarize in multiplicative notation some definitions and results from [KS05] and [Kuh00].