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Symmetric Attractors for Diffeomorphisms and Flows
Author(s) -
Field Michael,
Melbourne Ian,
Nicol Matthew
Publication year - 1996
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-72.3.657
Subject(s) - mathematics , equivariant map , attractor , invertible matrix , pure mathematics , symmetry (geometry) , axiom , group (periodic table) , reflection symmetry , diffeomorphism , symmetry group , dynamical systems theory , mathematical analysis , geometry , physics , quantum mechanics
Let Γ ⊂ O ( n ) be a finite group acting on R n . In this work we describe the possible symmetry groups that can occur for attractors of smooth (invertible) Γ‐equivariant dynamical systems. In case R n contains no reflection planes and n ⩾ 3, our results imply that there are no restrictions on symmetry groups. In case n ⩾ 4 (diffeomorphisms) and n ⩾ 5 (flows), we show that we may construct attractors which are Axiom A. We also give a complete description of what can happen in low dimensions.