3 ms·

multiplying by two, and adding I presume. makes sense. to give some related knowledge, all matrices can be decomposed into direct sums of shifts and scales. th

multiplying by two, and adding I presume. makes sense. to give some related knowledge, all matrices can be decomposed into direct sums of shifts and scales. this is the base of the jordan normal form [1]. infinite matrices can certainly encapsulate chaos. [1] http://en.wikipedia.org/wiki/Jordan_normal_form http://en.wikipedia.org/wiki/Jordan_normal_form

lotharbot 13y ago
There's actually no adding. Just multiplying by two. That's what makes the result so surprising. The key is the metric on the bi-infinite sequence space. Differing in the digit closest to the decimal point on either side gives a distance of 1/2, differing one digit further out is 1/4, then 1/8, 1/16, and so on (the total distance is the sum; the maximum distance between two sequences is 2 if they differ in every digit.) So two sequences are "close" to one another if their middle digits are all the same. This gives you the key attributes of chaos: (1) Sensitive Dependence on Initial Conditions. Two sequences can be identical for BIG_NUM digits on each side of the decimal point, and then completely different outside of that area. This means their initial distance apart is roughly 1/2^BIG_NUM. But after BIG_NUM multiply-by-two operations, the trajectories have diverged to a distance of around 2. (2) Topological Mixing. A similar construction to above -- select the first N digits of a sequence to make it fall into a particular neighborhood, and the next M digits to make it fall into a different neighborhood after the appropriate number of bitshifts. This can be done for any pair of neighborhoods. (3) Density of periodic orbits. Pick any point X in the space, and a distance epsilon. Construct a periodic orbit that gets within epsilon of X simply by taking the central digits of X (using log_2(epsilon) to select how many digits) and repeating those digits. There you have it -- chaos as a result of a single bitshift operation on bi-infinite sequences, with no addition or other operations.

lotharbot 13y ago

There's actually no adding. Just multiplying by two. That's what makes the result so surprising. The key is the metric on the bi-infinite sequence space. Differing in the digit closest to the decimal point on either side gives a distance of 1/2, differing one digit further out is 1/4, then 1/8, 1/16, and so on (the total distance is the sum; the maximum distance between two sequences is 2 if they differ in every digit.) So two sequences are "close" to one another if their middle digits are all the same. This gives you the key attributes of chaos: (1) Sensitive Dependence on Initial Conditions. Two sequences can be identical for BIG_NUM digits on each side of the decimal point, and then completely different outside of that area. This means their initial distance apart is roughly 1/2^BIG_NUM. But after BIG_NUM multiply-by-two operations, the trajectories have diverged to a distance of around 2. (2) Topological Mixing. A similar construction to above -- select the first N digits of a sequence to make it fall into a particular neighborhood, and the next M digits to make it fall into a different neighborhood after the appropriate number of bitshifts. This can be done for any pair of neighborhoods. (3) Density of periodic orbits. Pick any point X in the space, and a distance epsilon. Construct a periodic orbit that gets within epsilon of X simply by taking the central digits of X (using log_2(epsilon) to select how many digits) and repeating those digits. There you have it -- chaos as a result of a single bitshift operation on bi-infinite sequences, with no addition or other operations.