Fractional iteration of series and transseries (Q2849031)

Here the author investigates compositional iteration of fractional order for transseries. The main result of the paper is proving the existence of a family \(T^{[s]}\) indexed by real numbers \(s\) corresponding to the iteration of order \(s\) for any large positive transseries \(T\) of exponentiality \(0\) in the well-based case. Here, also an example of a grid-based large positive transseries \(T\) for which the fractional iterate \(T^{[s]}\) is not grid-based is presented. The author also studies the existence of a family \(T^{[s]}, s\in \mathbb{R}\), all supported by a single grid or by a single well-ordered set. He proves that such a family exists in both shallow and moderate cases but not in deep cases.