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Skew Products of Shifts with a Compact Lie Group
Author(s) -
Parry William
Publication year - 1997
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/s0024610797005462
Subject(s) - ergodic theory , mixing (physics) , mathematics , ergodicity , lie group , pure mathematics , skew , eigenfunction , aperiodic graph , constructive proof , group (periodic table) , constructive , mathematical analysis , combinatorics , physics , quantum mechanics , eigenvalues and eigenvectors , statistics , process (computing) , astronomy , computer science , operating system
We consider aperiodic shifts of finite type σ with an equilibrium state m and associated skew‐products σ f where f : X → G is Hölder and G is a compact Lie group. We show that generically σ f is weak‐mixing and give constructive methods for achieving weak‐mixing by perturbing an arbitrary f at a finite number of small neighbourhoods. When σ f is not ergodic we describe the (closed) ergodic decomposition precisely. The key result shows that certain measurable eigenfunctions are essentially Hölder continuous. This leads to conditions for weak‐mixing or ergodicity in terms of a functional equation which involves Hölder rather than measurable functions.