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Weak almost iterability. (English) Zbl 1187.26002

Let \(X\) be a real, compact interval. A continuous iteration semigroup is a continuous function \(F:(0,\infty) \times X \to X\) which satisfies the translation equation \(F(t,F(s,x))=F(s+t,x)\). An iterable function is a continuous function \(f:X\to X\) for which there is a continuous iteration semigroup \(F\) with \(F(n,x)=f^n(x)\) for all \(x\in X\) and \(n\in{\mathbb N}\). A weak almost iterable function is a continuous function \(f:X\to X\) for which there exists an iterable function \(g:X\to X\) such that \(\lim_{n\to\infty} (f^n(x)-g^n(x))=0\) for all \(x\in X\). In this paper, necessary and sufficient conditions are given for a function to be weak almost iterable. In the same way, functions are investigated for which the limit condition is only supposed to hold on a dense set, and functions for which the convergence holds with respect to certain Borel measures on \(X\).

MSC:

26A18 Iteration of real functions in one variable
39B12 Iteration theory, iterative and composite equations
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