Invariants of identity-tangent diffeomorphisms expanded as series of multitangents and multizetas

Abstract

We return to the subject of local, identity-tangent diffeomorphisms ƒ of ℂ and their analytic invariants A _ω (ƒ), under the complementary viewpoints of effective computation and explicit expansions. The latter rely on two basic ingredients: the so-called multizetas (transcendental numbers) and multitangents (transcendental functions), with resurgence monomials and their monics providing the link between the two. We also stress the difference between the collectors (preinvariant but of one piece) and the connectors (invariant but mutually unrelated).

Much attention has been paid to streamlining the nomenclature and notations. On the analysis side, resurgence theory rules the show. On the algebraic or combinatorial side, mould theory brings order and structure into the profusion of objects. Along the way, the paper introduces quite a few novel notions: new alien operators, new forms of resurgence, new symmetry types for moulds. It also broaches the subject of ‘phantom dynamics’ (dealing with formal diffeos that nonetheless possess invariants A _ω (ƒ)) and culminates in the comparison of arithmetical and dynamical monics, a distinction that reflects the dual nature of the A _ω (ƒ) as Stokes constants and holomorphic invariants.