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Invariant quadrics and orbits for a family of rational systems of difference equations. (English) Zbl 1302.39023

Summary: We study the existence of invariant quadrics for a class of systems of difference equations in \(\mathbb R^n\) defined by linear fractionals sharing denominator. Such systems can be described in terms of some square matrix \(A\) and we prove that there is a correspondence between non-degenerate invariant quadrics and solutions to a certain matrix equation involving \(A\). We show that if \(A\) is semisimple and the corresponding system admits non-degenerate quadrics, then every orbit of the dynamical system is contained either in an invariant affine variety or in an invariant quadric.

MSC:

39A22 Growth, boundedness, comparison of solutions to difference equations
15A69 Multilinear algebra, tensor calculus
15A18 Eigenvalues, singular values, and eigenvectors
15A63 Quadratic and bilinear forms, inner products
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