Arboreal models and their stability

TL;DR

This paper establishes the uniqueness and stability of arboreal models in symplectic topology, showing that local symplectic structures near these models are governed by combinatorial tree automorphisms.

Contribution

It proves the uniqueness of arboreal models and demonstrates that the local symplectic topology around them is determined by combinatorial tree automorphisms.

Findings

Uniqueness of arboreal models is established.

Local symplectic topology reduces to combinatorics.

Space of symplectomorphisms preserving a model is homotopy equivalent to tree automorphisms.

Abstract

This is the first in a series of papers by the authors on the arborealization program. The main goal of the paper is the proof of uniqueness of arboreal models, defined as the closure of the class of smooth germs of Lagrangian submanifolds under the operation of taking iterated transverse Liouville cones. The parametric version of the stability result implies that the space of germs of symplectomorphisms that preserve a canonical model is weakly homotopy equivalent to the space of automorphisms of the corresponding signed rooted tree. Hence the local symplectic topology around a canonical model reduces to combinatorics, even parametrically.