An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings

Abstract

A snapback repeller of an analytic mapping is defined as a full orbit which tends to an unstable fixed point backwards in time and snaps back to the same fixed point. This note gives a rather elementary proof that unstable periodic orbits accumulate near snapback repellers. The proof is entirely selfcontained and uses only standard elementary tools. We exploit that the global semiconjugacy of the entire analytic map to a linear map is itself an entire analytic function and apply the Theorem of Rouche to its zeros. We also generalize Marotto's result about the chaotic motion near a snapback repeller to include the degenerate case.