Equivariant embedding of finite-dimensional dynamical systems

We prove an equivariant version of the classical Menger-Nobeling 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) 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. 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 subset of G so that for a generic continuous map h : X -> [0, 1](r), the map h(F) : X -> ([0, 1](r))F given by x (sic) ( f (gx))(g is an element of F) is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.