[2.6] What is a map? A map is simply a function, f, on the phase space that gives the next state, f(z) (the image), of the system given its current state, z. (Often you will find the notation z' = f(z), where the prime means the next point, not the derivative.) Now a function must have a single value for each state, but there could be s everal different states that give rise to the same image. Maps that allow ev ery state in the phase space to be accessed (onto) and which have precisely one pre-image for each state (one-to-one) are invertible. If in addition the map and its inverse are continuous (with respect to the phase space coordin ate z), then it is called a homeomorphism. A homeomorphism that has at least one continuous derivative (w.r.t. z) and a continuously differentiable inve rse is a diffeomorphism. Iteration of a map means repeatedly applying the map to the consequents of t he previous application. Thus we get a sequence n z = f(z ) = f(f(z )...) = f (z ) n n-1 n-2 0 This sequence is the orbit or trajectory of the dynamical system with initia l condition z_0. -- 在細雨的午後 書頁裡悉哩哩地傳來 " 週期3 = ? " 然而我知道 當我正在日耳曼深處的黑森林 繼續發掘海森堡未曾做過的夢時 康德的諾言早已遠離......... 遠來的傳教士靜靜地看著山澗不斷反覆疊代自己的 過去 現在 和 未來 於是僅以 一顆量子渾沌 一本符號動力學 祝那發生在週一下午的新生 --

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