Processed Methods and Conjugacy

One of the intriguing ideas that can be exploited to improve numerical integrators is that of conjugacy. In general, we say that two maps A and B are conjugate if there is a homeomorphism x such that 

That is, we may evaluate one map by first transforming the input, then applying the other map, then transforming the output by the inverse of the first transformation. Conjugate maps have the property that their iterates are also conjugate, since 

If A and B are maps of phase space, then the conjugacy implies that they have equivalent stability properties under iteration, since if B (zo) z, as n oo, for all initial points zo, then also A (zo) / z ). 

Let us suppose that we have such a conjugacy between two numerical methods and h, that is 

The use of conjugacy as a method to develop enhanced numerical methods was introduced in [397] under the name symplectic correctors. 

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