Kick factorization of symplectic maps

There are two tacit assumptions made in the design and construction of accelerators and storage rings. The first, made by machine builders, is that if two machines are nearly the same, their performance (including long-term orbit stability) should be nearly the same. The second, made by accelerator theorists, is that analytical/numerical models of machine behavior, despite their approximate nature, still have relevance for real machines. From a mathematical perspective, what is being assumed in either case is that if two symplectic maps are close, then the behavior (including long-term behavior) of systems described by these maps should be nearly the same. The authors explore in a mathematical context, and for a simple example, some aspects of this assumption.<>