Equivariant embedding of finite-dimensional dynamical systems

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 ([0,1]ʳ)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 rN/2 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 hF:X→ ([0,1]ʳ)F h F : X → ( [ 0 , 1 ] r ) F given by x↦ (f(gx))g∈ F x ↦ ( f ( g x ) ) g ∈ F is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.