Cyclic Approximation of Irrational Rotations
نویسندگان
چکیده
منابع مشابه
Irrational rotations motivate measurable sets
In 1914 Constantin Carathéodory gave his definition of a measurable set, a definition that is crucial in the general theory of integration. This is because a “primitive” notion of area or measure, on a smaller family of sets, can be extended to the larger family of measurable sets, and it is this larger family that has the necessary properties for a natural and complete theory of integration. T...
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Let T = [0, 1) be the additive group of real numbers modulo 1, α ∈ T be an irrational number and t ∈ T. We consider skew product extensions of irrational rotations by Z 2 determined by T : T × Z 2 → T × Z 2 T (x, s 1 , s 2) = " x + α, s 1 + 2χ [0, 1 2) (x) − 1, s 2 + 2χ [0, 1 2) (x + t) − 1 «. We study ergodic components of such extensions and use the results to display irregularities in the un...
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We introduce a renormalization procedure which allows us to study in a unified and concise way different properties of the irrational rotations on the unit circle β 7→ {α+ β}, α ∈ R \ Q. In particular we obtain sharp results for the diffusion of the walk on Z generated by the location of points of the sequence {nα + β} on a binary partition of the unit interval. Finally we give some application...
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1 A 1 < £ < , 5B B SB so that we have the classical theorem of Hurwitz. For other values of r, approximations from both sides are permitted, but the errors allowed on the two sides are different; hence the term unsymmetrical approximation. The result here was new, and is so related to Hurwitz's inequality that one side is strengthened and the other weakened. Notice that the result for r > l is ...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1994
ISSN: 0002-9939
DOI: 10.2307/2160263