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An analogue to the Smale-Birkhoff homoclinic theorem for iterated entire mappings. (English) Zbl 0809.58033

There is considered a discrete dynamical system constructed by iterations of an entire analytic mapping \(f : \mathbb{C} \to \mathbb{C}\) having a hyperbolically unstable fixed point \(x = 0\). A snapback repeller is a full orbit coming from a fixed point \(x = 0\) and jumping back to the same fixed point.
The dynamics near a snapback repeller for nonanalytic smooth mappings from \(\mathbb{R}^ n\) to \(\mathbb{R}^ n\) was investigated by F. R. Marotto [J. Math. Anal. Appl. 63, 199-223 (1978; Zbl 0381.58004)] with a certain additional nondegeneracy assumption. There is given a new, quite elementary proof of that periodic orbits with high periods accumulate near a snapback repeller. The nondegeneracy assumption can be dropped.
Also Marotto’s result about the chaotic motion near a snapback repeller is generalized.

MSC:

37F99 Dynamical systems over complex numbers
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37G99 Local and nonlocal bifurcation theory for dynamical systems

Citations:

Zbl 0381.58004
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References:

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