Abstract
We are concerned with finitely nested square roots which are roots of iterations of a real quadratic polynomial $x^2-c$ with $c\geq 2$, and the limits of such nested square roots. We investigate how they are related to a Poincaré function $f(x)$ satisfying the functional equation $f(sx)=f(x)^2-c$, where $s=1+\sqrt{1+4c}$. Our main theorems can be viewed as a natural generalization of the work of Wiernsberger and Lebesgue for the case $c=2$. The key ingredients of the proof are some analytic properties of $F(x)$, which have been intensively studied by the second author using infinite compositions.
Citation
Noboru AOKI. Shota KOJIMA. "Nested Square Roots and Poincaré Functions." Tokyo J. Math. 39 (1) 241 - 269, June 2016. https://doi.org/10.3836/tjm/1471873313
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