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In the mathematical theory of dynamical systems, an irrational rotation is a map : where ''θ'' is an irrational number. Under the identification of a circle with R/Z, or with the interval (1 ) with the boundary points glued together, this map becomes a rotation of a circle by a proportion ''θ'' of a full revolution (i.e., an angle of 2''πθ'' radians). Since ''θ'' is irrational, the rotation has infinite order in the circle group and the map ''T''''θ'' has no periodic orbits. Alternatively, we can use multiplicative notation for an irrational rotation by introducing the map : The relationship between the additive and multiplicative notations is the group isomorphism :. It can be shown that is an isometry. There is a strong distinction in circle rotations that depends on whether is rational or irrational. Rational rotations are less interesting examples of dynamical systems because if and , then when . It can also be shown that when . == Significance == Irrational rotations form a fundamental example in the theory of dynamical systems. According to the Denjoy theorem, every orientation-preserving -diffeomorphism of the circle with an irrational rotation number is topologically conjugate to . An irrational rotation is a measure-preserving ergodic transformation, but it is not mixing. The Poincaré map for the dynamical system associated with the Kronecker foliation on a torus with angle is the irrational rotation by . C *-algebras associated with irrational rotations, known as irrational rotation algebras, have been extensively studied. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「irrational rotation」の詳細全文を読む スポンサード リンク
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