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Inexact orbits of holomorphic mappings in complex Banach spaces. (English) Zbl 1323.47059

This note contains two stability results on infinite-dimensional holomorphic dynamical systems.
Let \(\Omega\subset X\) be a bounded convex domain in a complex Banach space \(X\), and \(f:\Omega\to\Omega\) a holomorphic self-map of \(\Omega\). Assume that \(f(\Omega)\) is strictly inside \(\Omega\), that is, that the distance between \(f(\Omega)\) and \(\partial\Omega\) is strictly positive. Then it is well known that under these assumptions \(f\) has a unique fixed point \(z_f\in\Omega\), and that the orbit (under the action of \(f\)) of any point \(z_0\in\Omega\) converges to \(z_f\). Here the authors show that \(z_f\) is stable under perturbations on \(f\): for any \(\varepsilon>0\) there exists \(\delta>0\) such that any holomorphic self-map \(g:\Omega\to\Omega\) satisfying \(\|g(z)-f(z)\|\leq\delta\) for all \(z\in\Omega\) has a unique fixed point \(z_g\in\Omega\) and, moreover, \(\|z_g-z_f\|\leq\varepsilon\).
The authors also show that for any \(\varepsilon>0\) there exists \(\delta>0\) such that any \(\delta\)-pseudo-orbit \(\{z_j\}\) (i.e., \(\|z_{j+1}-f(z_j)\|\leq\delta\) for all \(j\geq 0\)) is eventually \(\varepsilon\)-close to \(z_f\), that is \(\|z_j-z_f\|\leq\varepsilon\) for all \(j\) large enough.
Reviewer: Marco Abate (Pisa)

MSC:

47H10 Fixed-point theorems
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37C20 Generic properties, structural stability of dynamical systems
46G20 Infinite-dimensional holomorphy
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
39B82 Stability, separation, extension, and related topics for functional equations
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References:

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