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Ergodicity for piecewise smooth cocycles over toral rotations

Anzelm Iwanik — 1998

Fundamenta Mathematicae

Let α be an ergodic rotation of the d-torus 𝕋 d = d / d . For any piecewise smooth function f : 𝕋 d with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on L 2 ( 𝕋 d ) is shown to have a continuous non-Dirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product S f : 𝕋 d + 1 𝕋 d + 1 must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum of V...

How restrictive is topological dynamics?

Anzelm Iwanik — 1997

Commentationes Mathematicae Universitatis Carolinae

Let T be a permutation of an abstract set X . In ZFC, we find a necessary and sufficient condition it terms of cardinalities of the T -orbits that allows us to topologize ( X , T ) as a topological dynamical system on a compact Hausdorff space. This extends an early result of H. de Vries concerning compact metric dynamical systems. An analogous result is obtained for 𝐙 2 -actions without periodic points.

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