# The Natural Growth Scale

• ### Jean Ecalle

Université Paris-Sud, Université de Paris-Saclay, Orsay, France

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## Abstract

The present paper starts with the group of all germs of analytic self-mappings of $\mathbb R_{,+\infty}$ and concerns itself with its successive closures under (i) fractional iteration (ii) conjugation (iii) the solving of general composition equations.

Rather than attempting a systematic treatment, we focus on the typical difficulties attendant upon these extensions. On the formal side, power series make way first for transseries, then for ultraseries, involving finite resp. transfinite iterates of the exponential. On the analysis side, the first casualties are convergence and analyticity: from the start, we have to face generic resurgence (multicritical but of a weakly polarising type) and, further down the road, generic cohesiveness (a natural and very inclusive extension of Denjoy quasi-analyticity).

Nevertheless, none of these complications destroys the bi-constructive correspondence between the formal objects (series, transseries, ultraseries) and the geometric germs. We describe, and illustrate on numerous examples, the apparatus required for upholding this correspondence: mainly accelero-summation, which uses convolution-respecting integral transforms to ascend from one critical Borel plane to the next, and the so-called display, a semi-algebraic construct that supplements the genuine variable with a host of pseudo-variables and encapsulates in highly convenient form all the information about the resurgence pattern and Stokes constants of a given germ.

We also devote three sections to the (non-linear) iso-differential operators which, on top of their surprising algebraic properties, are uniquely adapted to germ composition, the analysis of deep convexity, and the description of the universal asymptotics of very slow- or fast-growing germs.

Lastly, we reflect on the seemingly unsurmountable indeterminacy inherent in the choice of transfinite exponential iterates, and on the implications of that indeterminacy for the natural growth scale(by which we mean, roughly speaking, the ultimate extension of our groups of non-oscillating germs): far from being the quintessential continuum that one would expect, the natural growth scale – on the formal as on the analysis side, in the large as well as locally – displays a granular, almost fractal-like structure.