Form Symmetries and Reduction of Order in Difference Equations
presents a new approach to the formulation and analysis of
difference equations in which the underlying space is typically an
algebraic group. In some problems and applications, an additional
algebraic or topological structure is assumed in order to define
equations and obtain significant results about them. Reflecting the
author s past research experience, the majority of examples involve
equations in finite dimensional Euclidean spaces.
The book first introduces difference equations on groups,
building a foundation for later chapters and illustrating the wide
variety of possible formulations and interpretations of difference
equations that occur in concrete contexts. The author then proposes
a systematic method of decomposition for recursive difference
equations that uses a semiconjugate relation between maps. Focusing
on large classes of difference equations, he shows how to find the
semiconjugate relations and accompanying factorizations of two
difference equations with strictly lower orders. The final chapter
goes beyond semiconjugacy by extending the fundamental ideas based
on form symmetries to nonrecursive difference equations.
With numerous examples and exercises, this book is an ideal
introduction to an exciting new domain in the area of difference
equations. It takes a fresh and all-inclusive look at difference
equations and develops a systematic procedure for examining how
these equations are constructed and solved.
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