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Piecewise Absolutely Continuous Cocycles Over Irrational Rotations
Author(s) -
Iwanik A.,
Lemańczyk M.,
Mauduit C.
Publication year - 1999
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/s0024610799006961
Subject(s) - mathematics , absolute continuity , irrational number , piecewise , pure mathematics , discontinuity (linguistics) , mixing (physics) , lebesgue integration , spectrum (functional analysis) , mathematical analysis , operator (biology) , continuous function (set theory) , integer (computer science) , norm (philosophy) , function (biology) , combinatorics , physics , geometry , quantum mechanics , biochemistry , chemistry , repressor , evolutionary biology , biology , computer science , transcription factor , political science , law , gene , programming language
For an irrational rotation α of the circle group T = R / Z and a piecewise absolutely continuous function f : T → R , the unitary operator Vh ( x )=e 2πi f ( x ) h ( x +α) on L 2 ( T ) is studied. It is shown that if f has a single discontinuity with non‐integer jump then V is κ‐weakly mixing for some κ with 0<∣κ∣<1. In particular V has continuous singular spectrum. The property of κ‐weak mixing (with possible change of the value of κ, 0<∣κ∣<1) holds for all irrational rotations and, given α, is stable under perturbations of f by functions with sufficiently small O (1/ n )‐norm. On the other hand, there exists a piecewise linear function f with two non‐integer jumps such that the spectrum of V is continuous singular for one value of α and Lebesgue for another.