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:''This article is about the ''rotation number'', which is sometimes called the ''map winding number'' or simply ''winding number''. There is another meaning for winding number, which appears in complex analysis.'' In mathematics, the rotation number is an invariant of homeomorphisms of the circle. It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincaré later proved a theorem characterizing the existence of periodic orbits in terms of rationality of the rotation number. == Definition == Suppose that ''f'': ''S''1 → ''S''1 is an orientation preserving homeomorphism of the circle ''S''1 = R/Z. Then ''f'' may be lifted to a homeomorphism ''F'': R → R of the real line, satisfying : for every real number ''x'' and every integer ''m''. The rotation number of ''f'' is defined in terms of the iterates of ''F'': : Henri Poincaré proved that the limit exists and is independent of the choice of the starting point ''x''. The lift ''F'' is unique modulo integers, therefore the rotation number is a well-defined element of R/Z. Intuitively, it measures the average rotation angle along the orbits of ''f''. 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「Rotation number」の詳細全文を読む スポンサード リンク
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