Complex squaring map: Difference between revisions

Revision as of 14:15, 6 August 2009

Revision as of 22:33, 8 August 2009

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:<math> \qquad z_{n+1} = z_n^2 </math>

where <math>z_n</math> is the resulting sequence of complex numbers obtained by iterating the steps above, and <math>z_0</math> represents the initial starting number. We can solve this iteration exactly:

:<math> \qquad z_n = z_0^{2n2^n} </math>

Starting with angle θ, we can write the initial term as <math>z_0 = \exp(i\theta)</math> so that <math>z_n = \exp(i2ni2^n\theta)</math>. This makes the successive doubling of the angle clear. Also, the relation <math>z_n = \cos(2n2^n\theta)+i \sin(2n2^n\theta)</math> makes the wrapping of the angle onto the circle obvious. The doubling-and-wrapping is the same origin of chaos as the stretching-and-folding of the [[logistic map]], for example, and leads to exponential divergence of close starting angles (see [[Lyapunov exponent]]s). If the initial angle is expanded in a binary representation, then the iteration can be clearly seen to be equivalent to the [[2x mod 1 map|shift map]].

== Generalisations ==