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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Arnold's cat map ： ウィキペディア英語版 Arnold's cat map In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name.〔 Thinking of the torus $\backslash mathbb^2$ as the quotient space $\backslash mathbb^2/\backslash mathbb^2$ Arnold's cat map is the transformation $\backslash Gamma\; :\; \backslash mathbb^2\; \backslash to\; \backslash mathbb^2$ given by the formula :$\backslash Gamma\; \backslash ,\; :\; \backslash ,\; (x,y)\; \backslash to\; (2x+y,x+y)\; \backslash bmod\; 1.$ Equivalently, in matrix notation, this is : That is, with a unit size equal to the width of the square image, the image is sheared one unit up, then one unit to the right, and all that lies outside that unit square is shifted back by the unit until it's within the square. ==Properties== * Γ is invertible because the matrix has determinant 1 and therefore its inverse has integer entries, * Γ is area preserving, * Γ has a unique hyperbolic fixed point (the vertices of the square). The linear transformation which defines the map is hyperbolic: its eigenvalues are irrational numbers, one greater and the other smaller than 1 (in absolute value), so they are associated respectively to an expanding and a contracting eigenspace which are also the stable and unstable manifolds. The eigenspace are orthogonal because the matrix is symmetric. Since the eigenvectors have rationally independent components both the eigenspaces densely cover the torus. Arnold's cat map is a particularly well-known example of a ''hyperbolic toral automorphism'', which is an automorphism of a torus given by a square unimodular matrix having no eigenvalues of absolute value 1.〔 * The set of the points with a periodic orbit is dense on the torus. Actually a point is preperiodic if and only if its coordinates are rational. * Γ is topologically transitive (i.e. there is a point whose orbit is dense, this happens for example for any points on the expanding eigenspace) * The number of points with period ''n'' is exactly |λ1''n'' + λ2''n''−2| (where λ1 and λ2 are the eigenvalues of the matrix). For example, the first few terms of this series are 1, 5, 16, 45, 121, 320, 841, 2205 .... (The same equation holds for any unimodular hyperbolic toral automorphism if the eigenvalues are replaced.) * Γ is ergodic and mixing, * Γ is an Anosov diffeomorphism and in particular it is structurally stable. 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「Arnold's cat map」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.