Some Periodic and Non-Periodic Recursions

Different Approaches to the Global Periodicity Problem

“…The piecewise linear map F(x, y) = (y, max(0, y) − x) is 5-periodic, see [20]. Its associated globally 5-periodic recurrence is…”

Normal forms for rational difference equations with applications to the global periodicity problem

“…It is well known that the condition A p = Id is equivalent to the fact that A is a diagonalizable matrix and its eigenvalues are p-roots of the unity, in particular they lie on the unit circle. It is proved in [12], that when F is of the form F (x) = (x 2 , . .…”

“…To our knowledge, all the known periodic rational difference equations of type (1) satisfying (I)-(III) (modulo a conjugation of their associated maps via an homothety, that is, an equivalence of the equations in the terminology used in [9,12]) are:…”

“…The study of globally periodic difference equations is a current subject of research, (see for instance [1][2][3][4][5][6][7]9,11,12,14,16,19,21,22]), and it is a part of a classical problem of the theory of functional equations which is the resolution of the Babbage equation F p = Id, see for instance [15] and references therein.…”

“…The complete characterization of periodic rational difference equations of order k (either with real or complex coefficients) is still an open problem ( [9,12,16] or [17]). To our knowledge, all the known periodic rational difference equations of type (1) satisfying (I)-(III) (modulo a conjugation of their associated maps via an homothety, that is, an equivalence of the equations in the terminology used in [9,12]) are:…”