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Ergodicity and mixing of non‐commuting epimorphisms
Author(s) -
Bergelson Vitaly,
Gorodnik Alexander
Publication year - 2007
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/pdm007
Subject(s) - mathematics , mixing (physics) , automorphism , abelian group , ergodic theory , irreducibility , ergodicity , pure mathematics , bijection, injection and surjection , endomorphism , order (exchange) , cardinality (data modeling) , combinatorics , physics , bijection , statistics , finance , quantum mechanics , computer science , economics , data mining
We study mixing properties of epimorphisms of a compact connected finite‐dimensional abelian group X . In particular, we show that a set F , with | F | > dim X , of epimorphisms of X is mixing if and only if every subset of F of cardinality (dim X ) + 1 is mixing. We also construct examples of free non‐abelian groups of automorphisms of tori which are mixing, but not mixing of order 3, and show that, under some irreducibility assumptions, ergodic groups of automorphisms contain mixing subgroups and free non‐abelian mixing subsemigroups.