Equivariant embedding of finite-dimensional dynamical systems

Abstract We prove an equivariant version of the classical Menger-Nöbeling theorem regarding topological embeddings: Whenever a group G acts on a finite-dimensional compact metric space X, a generic continuous equivariant function from X into (0, 1ʳ) G (0, 1 r) G is a topological embedding, provided that for every positive integer N the space of points in X with orbit size at most N has topological dimension strictly less than rN2 rN 2. We emphasize that the result imposes no restrictions whatsoever on the acting group G (beyond the existence of an action on a finite-dimensional space). Moreover if G is finitely generated then there exists a finite subset F G F ⊂ G so that for a generic continuous map h: X 0, 1^r h: X → 0, 1 r, the map h^F: X (0, 1^r) ^F h F: X → (0, 1 r) F given by x (f (gx) ) ₆ ₅ x ↦ (f (g x) ) g ∈ F is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.