kappa-bounded Exponential-Logarithmic Power Series Fields

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In math.AC/9608214 it was shown that fields of generalized power series cannot admit an exponential function. In this paper, we construct fields of generalized power series with bounded support which admit an exponential. We give a natural definition of an exponential, which makes these fields into models of real exponentiation. The method allows to construct for every kappa regular uncountable cardinal, 2kappa pairwise non-isomorphic models of real exponentiation (of cardinality kappa), but all isomorphic as ordered fields. Indeed, the 2kappa exponentials constructed have pairwise distinct growth rates. This method relies on constructing lexicographic chains with many automorphisms.