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Symbolic orbits of the tent map, 640x480 pixels.

In mathematics, and in particular, in the theory of dynamical systems, one has the concept of the symbolic orbit of a point. This graph illustrates the symbolic orbit of the unit-height tent map, expressed as a binary value.

The unit-height tent map is given by the equation:

${\displaystyle S_{\mathrm {tent} }(x)={\begin{cases}2x&{\mbox{for }}0\leq x<{\frac {1}{2}}\\2-2x&{\mbox{for }}{\frac {1}{2}}\leq x<1\end{cases}}}$

Picking an initial value for x, and iterating on this function, one gets the orbit for the point x. This orbit can be represented symbolically as a string of letters L and R, indicating whether the nth iterate fell to the left or to the right of 1/2. By the xxx theorem, the symbolic orbits are in one-to-one correspondance with the initial value x.

Since the orbit string consists only of two symbols, these two symbols can be taken to be the binary values 0 and 1; thus, the symbolic orbit string can also be interpreted as a real number expressed in binary. This graph shows the value of this orbit string as a function of x.

More precisely, let:

${\displaystyle s_{n+1}=S_{\mathrm {tent} }(s_{n})}$

with ${\displaystyle s_{0}=x}$, so that the ${\displaystyle s_{n}}$ are the iterates of the tent map. The let

${\displaystyle b_{n}={\begin{cases}0&{\mbox{for }}x<1/2\\1&{\mbox{for }}x\geq 1/2\end{cases}}}$

Finally, let:

${\displaystyle T(x)=\sum _{n=0}^{\infty }b_{n}/2^{n}}$

The graph shown is a graph of T(x).

Note that this mapping is only possible for tent maps of unit height or less. Taller maps will have orbits that require more than two symbols to express.

Created by Linas Vepstas User:Linas <linas@linas.org> on 25 June 2005 using custom software written entirely by Linas Vepstas.

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• (del) (cur) 17:39, 25 June 2005 . . en:User:Linas Linas ( en:User_talk:Linas Talk) . . 640x480 (5292 bytes) (Binary value of dyadic string of tent map orbits.)

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