Most monothetic extensions are rank-1
1993; Polish Academy of Sciences; Volume: 66; Issue: 1 Linguagem: Inglês
10.4064/cm-66-1-63-76
ISSN1730-6302
Autores Tópico(s)Advanced Topics in Algebra
ResumoIntroduction.Let T be an ergodic automorphism of a standard probability space (X, B, µ) and G be a compact metrizable abelian group.For any measurable mapping φ : X → G (a cocycle) we define an automorphismof X × G, called a G-extension of T .The investigation of ergodic properties of such skew products goes back to Anzai [A] who studied the case of X = G = T, the circle group, with T an irrational rotation.In [R2], E. A. Robinson, Jr. proved that typically the G-extensions have simple spectrum.More specifically, if T admits a "good cyclic approximation" then most (in the sense of category for the L 1 -distance in the space of cocycles) G-extensions have simple spectrum.In Section 2 of the present paper we show that if G is a monothetic group and T admits a cyclic approximation with speed o(1/n), a condition implied by the existence of a "good cyclic approximation", then most G-extensions are in fact rank-1.(Recall that rank-1 implies simple spectrum by Baxter [Ba].)In particular, if T z = e 2πiα z is an irrational rotation where α has unbounded partial quotients then most Anzai extensions of T are rank-1.Note that the set of such α's is large in the sense of both measure and category.It is well known that any discrete spectrum ergodic automorphism is rank-1 (see [J]).To make sure that the discrete spectrum extensions are not generic we prove in Section 3 that in fact a typical G-extension of any ergodic T has no eigenfunctions other than those of T .In other words, a generic cocycle is weakly mixing (Theorem 2).This extends an old result of Jones and Parry [J-P] where the same is proved assuming T to be weakly mixing.In particular, we may now conclude that a typical Anzai cocycle is both weakly mixing and rank-1.In Section 4 we focus on continuous Anzai cocycles φ : T → T of topological degree zero.Such cocycles play an important role in the theory of
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