. . . "1065282152"^^ . . "In the mathematical theory of dynamical systems, an irrational rotation is a map where \u03B8 is an irrational number. Under the identification of a circle with R/Z, or with the interval [0, 1] with the boundary points glued together, this map becomes a rotation of a circle by a proportion \u03B8 of a full revolution (i.e., an angle of 2\u03C0\u03B8 radians). Since \u03B8 is irrational, the rotation has infinite order in the circle group and the map T\u03B8 has no periodic orbits. Alternatively, we can use multiplicative notation for an irrational rotation by introducing the map . It can be shown that \u03C6 is an isometry."@en . . . . . "En teor\u00EDa matem\u00E1tica de los sistemas din\u00E1micos, una rotaci\u00F3n irracional es una funci\u00F3n matem\u00E1tica. donde \u03B8 es un n\u00FAmero irracional. En virtud de la identificaci\u00F3n de una circunferencia con R/Z o el intervalo [0, 1] con los puntos de los l\u00EDmites relacionados directamente entre s\u00ED, esta funci\u00F3n se convierte b\u00E1sicamente en una rotaci\u00F3n de una circunferencia en una proporci\u00F3n \u03B8 de una revoluci\u00F3n completa, es decir, un \u00E1ngulo de 2\u03C0\u03B8 radianes. Siendo \u03B8 irracional, la rotaci\u00F3n tiene orden infinito en el grupo circular, mientras que la funci\u00F3n T\u03B8 no tiene \u00F3rbitas peri\u00F3dicas. ."@es . . . . . . "Rotaci\u00F3n irracional"@es . . "En teor\u00EDa matem\u00E1tica de los sistemas din\u00E1micos, una rotaci\u00F3n irracional es una funci\u00F3n matem\u00E1tica. donde \u03B8 es un n\u00FAmero irracional. En virtud de la identificaci\u00F3n de una circunferencia con R/Z o el intervalo [0, 1] con los puntos de los l\u00EDmites relacionados directamente entre s\u00ED, esta funci\u00F3n se convierte b\u00E1sicamente en una rotaci\u00F3n de una circunferencia en una proporci\u00F3n \u03B8 de una revoluci\u00F3n completa, es decir, un \u00E1ngulo de 2\u03C0\u03B8 radianes. Siendo \u03B8 irracional, la rotaci\u00F3n tiene orden infinito en el grupo circular, mientras que la funci\u00F3n T\u03B8 no tiene \u00F3rbitas peri\u00F3dicas. Alternativamente, se puede usar la notaci\u00F3n multiplicativa para una rotaci\u00F3n irracional introduciendo la siguiente funci\u00F3n: La relaci\u00F3n entre las notaciones aditivas y multiplicativas es el isomorfismo: . Con esto se demuestra que \u03C6 es una isometr\u00EDa."@es . . . . "In the mathematical theory of dynamical systems, an irrational rotation is a map where \u03B8 is an irrational number. Under the identification of a circle with R/Z, or with the interval [0, 1] with the boundary points glued together, this map becomes a rotation of a circle by a proportion \u03B8 of a full revolution (i.e., an angle of 2\u03C0\u03B8 radians). Since \u03B8 is irrational, the rotation has infinite order in the circle group and the map T\u03B8 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 \u03C6 is an isometry. There is a strong distinction in circle rotations that depends on whether \u03B8 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 ."@en . . "9445837"^^ . . . . "6944"^^ . . . . . . . . . . . . . . . . . . . "Irrational rotation"@en . . . . . . . "\u529B\u5B66\u7CFB\u306E\u6570\u5B66\u7406\u8AD6\u306B\u304A\u3044\u3066\u3001\u7121\u7406\u56DE\u8EE2\uFF08\u3080\u308A\u304B\u3044\u3066\u3093\u3001\u82F1: irrational rotation\uFF09\u3068\u306F\u3001\u6B21\u306E\u5199\u50CF\u306E\u3053\u3068\u3092\u8A00\u3046\uFF1A \u4F46\u3057 \u03B8 \u306F\u7121\u7406\u6570\u3067\u3042\u308B\u3002\u5186\u3092 R/Z\u3001\u3042\u308B\u3044\u306F\u5883\u754C\u304C\u8CBC\u308A\u5408\u308F\u3055\u308C\u308B\u533A\u9593 [0, 1] \u3068\u898B\u306A\u3059\u3068\u3001\u3053\u306E\u5199\u50CF\u306F\u5168\u56DE\u8EE2\u306B\u5BFE\u3059\u308B\u5272\u5408 \u03B8\uFF08\u3059\u306A\u308F\u3061\u30012\u03C0\u03B8 \u30E9\u30B8\u30A2\u30F3\u306E\u3042\u308B\u89D2\uFF09\u306B\u3088\u308B\u5186\u306E\u56DE\u8EE2\u3092\u8868\u3059\u3053\u3068\u306B\u306A\u308B\u3002\u03B8 \u306F\u7121\u7406\u6570\u3067\u3042\u308B\u306E\u3067\u3001\u3053\u306E\u56DE\u8EE2\u306F\u5186\u5468\u7FA4\u306B\u304A\u3044\u3066\u7121\u9650\u306E\u4F4D\u6570\u3092\u6301\u3061\u3001\u5199\u50CF T\u03B8 \u306F\u5468\u671F\u8ECC\u9053\u3092\u6301\u305F\u306A\u3044\u3002 \u4E0A\u306E\u4EE3\u308F\u308A\u306B\u3001\u7121\u7406\u56DE\u8EE2\u306F\u4E57\u6CD5\u3092\u7528\u3044\u3066\u6B21\u306E\u5199\u50CF\u306E\u3088\u3046\u306B\u8868\u3059\u3053\u3068\u3082\u51FA\u6765\u308B\uFF1A \u3053\u308C\u3089\u52A0\u6CD5\u3068\u4E57\u6CD5\u306E\u8A18\u6CD5\u306E\u9593\u306B\u3042\u308B\u95A2\u4FC2\u306F\u3001\u7FA4\u540C\u578B . \u3067\u3042\u308B\u3002\u03C6 \u306F\u7B49\u9577\u3067\u3042\u308B\u3053\u3068\u3092\u793A\u3059\u3053\u3068\u3082\u51FA\u6765\u308B\u3002 \u03B8 \u304C\u6709\u7406\u6570\u3067\u3042\u308B\u304B\u7121\u7406\u6570\u3067\u3042\u308B\u304B\u306B\u5FDC\u3058\u3066\u3001\u5186\u5468\u306E\u56DE\u8EE2\u306B\u306F\u660E\u78BA\u306A\u533A\u5225\u304C\u5B58\u5728\u3059\u308B\u3002\u6709\u7406\u56DE\u8EE2\u306F\u3001 \u304A\u3088\u3073 \u3067\u3042\u308C\u3070 \u306B\u5BFE\u3057\u3066 \u306B\u306A\u308B\u3068\u3044\u3046\u4E8B\u5B9F\u3088\u308A\u3001\u529B\u5B66\u7CFB\u306B\u304A\u3044\u3066\u7121\u7406\u56DE\u8EE2\u307B\u3069\u306E\u8208\u5473\u3092\u5F15\u304F\u3082\u306E\u3067\u306F\u306A\u3044\u3002 \u3067\u3042\u308C\u3070 \u3092\u793A\u3059\u3053\u3068\u3082\u51FA\u6765\u308B\u3002"@ja . . . . . . . . . . . . . . . . "\u7121\u7406\u56DE\u8EE2"@ja . . . "\u529B\u5B66\u7CFB\u306E\u6570\u5B66\u7406\u8AD6\u306B\u304A\u3044\u3066\u3001\u7121\u7406\u56DE\u8EE2\uFF08\u3080\u308A\u304B\u3044\u3066\u3093\u3001\u82F1: irrational rotation\uFF09\u3068\u306F\u3001\u6B21\u306E\u5199\u50CF\u306E\u3053\u3068\u3092\u8A00\u3046\uFF1A \u4F46\u3057 \u03B8 \u306F\u7121\u7406\u6570\u3067\u3042\u308B\u3002\u5186\u3092 R/Z\u3001\u3042\u308B\u3044\u306F\u5883\u754C\u304C\u8CBC\u308A\u5408\u308F\u3055\u308C\u308B\u533A\u9593 [0, 1] \u3068\u898B\u306A\u3059\u3068\u3001\u3053\u306E\u5199\u50CF\u306F\u5168\u56DE\u8EE2\u306B\u5BFE\u3059\u308B\u5272\u5408 \u03B8\uFF08\u3059\u306A\u308F\u3061\u30012\u03C0\u03B8 \u30E9\u30B8\u30A2\u30F3\u306E\u3042\u308B\u89D2\uFF09\u306B\u3088\u308B\u5186\u306E\u56DE\u8EE2\u3092\u8868\u3059\u3053\u3068\u306B\u306A\u308B\u3002\u03B8 \u306F\u7121\u7406\u6570\u3067\u3042\u308B\u306E\u3067\u3001\u3053\u306E\u56DE\u8EE2\u306F\u5186\u5468\u7FA4\u306B\u304A\u3044\u3066\u7121\u9650\u306E\u4F4D\u6570\u3092\u6301\u3061\u3001\u5199\u50CF T\u03B8 \u306F\u5468\u671F\u8ECC\u9053\u3092\u6301\u305F\u306A\u3044\u3002 \u4E0A\u306E\u4EE3\u308F\u308A\u306B\u3001\u7121\u7406\u56DE\u8EE2\u306F\u4E57\u6CD5\u3092\u7528\u3044\u3066\u6B21\u306E\u5199\u50CF\u306E\u3088\u3046\u306B\u8868\u3059\u3053\u3068\u3082\u51FA\u6765\u308B\uFF1A \u3053\u308C\u3089\u52A0\u6CD5\u3068\u4E57\u6CD5\u306E\u8A18\u6CD5\u306E\u9593\u306B\u3042\u308B\u95A2\u4FC2\u306F\u3001\u7FA4\u540C\u578B . \u3067\u3042\u308B\u3002\u03C6 \u306F\u7B49\u9577\u3067\u3042\u308B\u3053\u3068\u3092\u793A\u3059\u3053\u3068\u3082\u51FA\u6765\u308B\u3002 \u03B8 \u304C\u6709\u7406\u6570\u3067\u3042\u308B\u304B\u7121\u7406\u6570\u3067\u3042\u308B\u304B\u306B\u5FDC\u3058\u3066\u3001\u5186\u5468\u306E\u56DE\u8EE2\u306B\u306F\u660E\u78BA\u306A\u533A\u5225\u304C\u5B58\u5728\u3059\u308B\u3002\u6709\u7406\u56DE\u8EE2\u306F\u3001 \u304A\u3088\u3073 \u3067\u3042\u308C\u3070 \u306B\u5BFE\u3057\u3066 \u306B\u306A\u308B\u3068\u3044\u3046\u4E8B\u5B9F\u3088\u308A\u3001\u529B\u5B66\u7CFB\u306B\u304A\u3044\u3066\u7121\u7406\u56DE\u8EE2\u307B\u3069\u306E\u8208\u5473\u3092\u5F15\u304F\u3082\u306E\u3067\u306F\u306A\u3044\u3002 \u3067\u3042\u308C\u3070 \u3092\u793A\u3059\u3053\u3068\u3082\u51FA\u6765\u308B\u3002"@ja .