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This is an attempt to expand the examples section of Symbolic dynamics. The current article shows nothing about the use of this technique.

Examples
Concepts such as heteroclinic orbits and homoclinic orbits have a particularly simple representation in symbolic dynamics.

The 2x modulo 1 map
The map 2x modulo 1 can be represented as an operation on a binary value. Using this representation, some properties of the map can be deduced.

Any value can be written as a sum of fractional binary digits $$d_n = \frac{1}{2^n}$$. All digits with n<1 represent a value greater than or equal to one, and by virtue of the modulo operation in the map they can be ignored. They will always be 0.

To apply the 2x modulo 1 map to this representation, all digits are shifted to the left, and the leftmost digit (that now represents the value 1) is dropped. From the values of the individual digits it is easy to see that the left-shifting of the digits in essence amounts to a multiplication by 2, while the removal of the leftmost digit, which has the value 1, corresponds with the modulo operation of the map.

Conclusions that can be gathered:
 * 1) With every aplication of the map, the digits will shift left, so digits far to the right (representing very small values), will eventually attain higher values. So two initial conditions separated by a small value will eventually diverge.
 * 2) If the binary representation repeats after a number of digits k, the same value for x will occur after every k applications of the map. This implies that all repeated binary fractions of length k are on period k orbits. So to find a period k orbit, it suffices to find or create a repeating pattern of length k. <>
 * 3) As there are only 2k possible permutations of k bits, this is also the number of orbits with period k.
 * 4) The modulo 1 operation drops a digit, which makes it impossible to invert the operation. So the map is non-invertable.