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Ergodicity of a Class of Cocycles Over Irrational Rotations
Author(s) -
Lemańczyk Mariusz,
Mauduit Christian
Publication year - 1994
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/49.1.124
Subject(s) - irrational number , ergodic theory , ergodicity , mathematics , skew , pure mathematics , rigidity (electromagnetism) , class (philosophy) , fourier series , fourier transform , product (mathematics) , crossed product , combinatorics , mathematical analysis , physics , geometry , algebra over a field , computer science , quantum mechanics , statistics , astronomy , artificial intelligence
It is proved that if α is irrational and Φ̃ ∈ L 2 ( S 1 ) with Φ̃ ∈o(l/ n ) then for each m ∈ Z \{0} the corresponding skew product ( e 2 π ix , e 2 π iy ) ↦ ( e 2 π i ( x + α ) , e 2 π i ( Φ ( x ) + mx + y ) ) is ergodic. The rigidity of special flows over irrational rotations with roof functions whose Fourier coefficients are in o(l/ n ) is also shown.