Tokyo Journal of Mathematics

Nested Square Roots and Poincaré Functions

Noboru AOKI and Shota KOJIMA

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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.

Article information

Tokyo J. Math., Volume 39, Number 1 (2016), 241-269.

First available in Project Euclid: 22 August 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 97I70: Functional equations
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 33B10: Exponential and trigonometric functions


AOKI, Noboru; KOJIMA, Shota. Nested Square Roots and Poincaré Functions. Tokyo J. Math. 39 (2016), no. 1, 241--269. doi:10.3836/tjm/1471873313.

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