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Convergence of a series whose terms are iterates of quadratic maps. (English) Zbl 1082.39017

The authors study the continuity of solutions of the functional equation \[ k(p(x))+k(x)=x, \] where \(p(x)=x^2+c\). The continuity of \(k\) depends on the convergence of the series \[ S(x):=\sum_{i=0}^{\infty} (p^{(2i)}(x)-p^{(2i+1)}(x)). \] Using discriminants and resultans the authors prove that in case \(c=-\frac{3}{4}\) the series \(S(x)\) is divergent for all \(x\in [-\frac{3}{4}, -\frac{1}{2})\cup (-\frac{1}{2},0]\). It implies that in this case there are no solutions of the considered functional equation which are continuous at \(x=-\frac{1}{2}\). For the cases where \(c=0\) and \(-\frac{3}{4}<c<0\) it is considered the problem whether the function \(S\) can be continuously extended from the maximal open interval on which the series is convergent to the boundary point of the interval.

MSC:

39B12 Iteration theory, iterative and composite equations
39B22 Functional equations for real functions
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