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Generic smooth cocycles of degree zero over irrational rotations

100%

Iwanik A.

Studia Mathematica

1995

tom 115

nr 3

241-250

If a rotation α of 𝕋 has unbounded partial quotients then "most" of its skew-product diffeomorphic extensions to the 2-torus 𝕋 × 𝕋 defined by $C^1$ cocycles of topological degree zero enjoy nontrivial ergodic properties. In fact they admit a cyclic approximation with speed o(1/n) and have nondiscrete (simple) spectrum. Similar results are obtained for $C^r$ cocycles if α admits a sufficiently good approximation by rationals. For a.e. α and generic $C^1$ cocycles the speed can be improved to o(1/(nlogn)). For generic α and generic $C^r$ cocycles (r = 1,...,∞) the spectral measure of the skew product has a continuous component and Hausdorff dimension zero.

Shift invariant measures and simple spectrum

88%

Kłopotowski A., Nadkarni M. G.

Colloquium Mathematicum

2000

tom 84/85

nr 2

385-394

We consider some descriptive properties of supports of shift invariant measures on $ℂ^{ℤ}$ under the assumption that the closed linear span (in $L^{2}$) of the co-ordinate functions on $ℂ^{ℤ}$ is all of $L^{2}$.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Studia Mathematica

1 Iwanik A.

1 Kłopotowski A.

1 Nadkarni M. G.

1 2000

1 1995

Wyniki wyszukiwania

Generic smooth cocycles of degree zero over irrational rotations

Shift invariant measures and simple spectrum