[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.
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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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