Global first integrals for first order difference equations

Abstract

Consider the autonomous, first order difference equation where the function f is analytic on its domain . Assume that this difference equation has at least one fixed point . A global first integral for this difference equation is an analytic function . If any of the fixed points are hyperbolic, then the only global first integral is the identically constant fuction. A global first integral exists if and only if the difference equation is invariant with respect to a one parameter Lie group of transformations. Existence of a nonconstant global first integral implies all the fixed points of the difference equation are nonhyperbolic.

Keywords:

• Difference equations
• First integrals
• Lie group transformations

AMS Subject Classifications:

• 22E
• 30D
• 39A
• 39B
• 54H

^∗Tel:(505)[email protected]

^∗Tel:(505)[email protected]

Notes

^∗Tel:(505)[email protected]