Equivariant embedding of finite-dimensional dynamical systems

2023/05/28 by Gutman, Yonatan, Levin, Michael, Meyerovitch, Tom
#37B99 #57M60 #Dynamical Systems (math.DS) #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2305.17717

Abstract

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⊂ 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↦ (f(gx))_{g∈ F} is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.

Related

• Equivariant embeddings of manifolds into Euclidean spaces
• Takens' embedding theorem with a continuous observable
• Equivariant geometry of Banach spaces and topological groups
• Finite-to-one equivariant maps and mean dimension
• Properly discontinuous actions versus uniform embeddings