TY - JOUR

T1 - Equivariant embedding of finite-dimensional dynamical systems

AU - Gutman, Yonatan

AU - Levin, Michael

AU - Meyerovitch, Tom

N1 - Publisher Copyright:
© The Author(s) 2024.

PY - 2024/1/1

Y1 - 2024/1/1

N2 - 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 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. 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 hF: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.

AB - 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 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. 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 hF: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.

UR - http://www.scopus.com/inward/record.url?scp=85198832519&partnerID=8YFLogxK

U2 - 10.1007/s00208-024-02911-y

DO - 10.1007/s00208-024-02911-y

M3 - Article

AN - SCOPUS:85198832519

SN - 0025-5831

JO - Mathematische Annalen

JF - Mathematische Annalen

ER -