Complex squaring map: Difference between revisions

Revision as of 12:41, 29 October 2009

Revision as of 12:42, 29 October 2009

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== Generalisations ==

This map is a special case of the complex [[quadratic map]], which has exact solutions for many special cases.<ref>M. Little, D. Heesch (2004), [http://www.eng.ox.ac.uk/samp/members/publications/GDEA41040.pdf Chaotic root-finding for a small class of polynomials], ''Journal of Difference Equations and Applications'', '''10'''(11):949&ndash;953.</ref> The complex map obtained by raising the previous number to any natural number power <math>z_{n+1} = z_n^p </math> is also exactly solvable as <math>z_n = z_0^{p^n}</math>. In the case ''p''&nbsp;=&nbsp;2, the dynamics can be mapped to the binarydyadic [[2x mod 1 map|shift map]]transformation, as described above, but for ''p''&nbsp;>&nbsp;2, we obtain a shift map in the [[number base]]&nbsp;''p''. For example, ''p''&nbsp;=&nbsp;10 is a decimal shift map.

== See also ==