I.e. does the parameter plane explain real dynamics? Real dynamics of iterative methods are not studied briefly due to the (wrong?) belief that real dynamics are included in complex dynamics.
My first answer a few years ago would be that of course the complex dynamics includes real dynamics but not I can say it is not true at all. For example, in real dynamics one can prove the monotony of sequences, etc.
Dear Ángel,
I think that the complex dynamics not necceary include real dynamics. Effects that appears with the phase are not reflected in the real number set.
I.E. the quadratic map has a clear interpretation used with real numbers, but the quadratic map in the complex plane, as sure you know, generate the families of Julia or Mandelbrot sets.
In summary the answer depends in the way in wich you do define the recursive iteration scheme.
Dear Walter,
That is exactly what I thought, because I hace found some situation in which the complex dynamics "lies" about the real dynamics but when I sent to a Journal referees said that real dynamics is included in complex ones.
Thanks a lot.
I regret the referees answer, but i remain in the same proposal with you !!!.
Real dynamics are most certainly addressed by several talented analysts and dynamicysts. See http://www.math.sunysb.edu/~mlyubich/papers/ICM.pdf, or http://abel.math.harvard.edu/archive/101_spring_00/www/papers/home/text/papers/real/book.pdf where a whole chapter is dedicated to real quadratic dynamics. Of course, there is Beardon's Iteration of Rational Functions: Complex Analytic Dynamical Systems, and our demigod Milnor and his Dynamics Text (best thing ever written).. still available for free at http://arxiv.org/pdf/math/9201272.pdf (he addresses aspects of real dynamics as well). But I, as a complex analyst, was always under the impression that real dynamics, or fluid dynamics, and ergodic theory all, in their own right, well-developed and rich fields. And to answer, your most immediate question, that the parameter space of a family of complex functions most certainly carries information about the real subset of said family. Iterative processes, even when limited to the simple real monic quadratic get really messy really fast. One of my advisors (from back in the day) Saeed Zakeri has a paper performing exactly what you ask, - he utilizes complex dynamics to arrive at theorems about real dynamics, see http://arxiv.org/pdf/math/0210382v1.pdf Hope I could help.
Dear Ross Flek, Thanks a lot for your response. I am glad to hear from people interested in real and complex analyst.
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