## Tokyo Journal of Mathematics

### Nested Square Roots and Poincaré Functions

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 22 August 2016

https://projecteuclid.org/euclid.tjm/1471873313

Digital Object Identifier
doi:10.3836/tjm/1471873313

Mathematical Reviews number (MathSciNet)
MR3543142

Zentralblatt MATH identifier
1360.39011

#### Citation

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

#### References

• M. Aschenbrenner and W. Bergweller, Julia's equation and differential transcendence, arXiv:1307.6381
• G. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles, Ann. Sci. Ec. Norm. Sup. (3) 1 (1884), Supplement, 3–41.
• S. Kojima, A generalization of trigonometric functions by infinite compositions of functions (in Japanese), Master Thesis at Rikkyo Univerity 2009.
• S. Kojima, On the infinite compositions of functions, Doctor Thesis at Rikkyo University 2011.
• S. Kojima, On the convergence of infinite compositions of entire functions, Arch. Math. 98 (2012), 453–465.
• H. Lebesgue, Sur certaines expressions irrationelles illimités, Bull. Calcatta Math. Soc. 29 (1937), 17–28.
• H. Lebesgue, Sur certaines expressions irrationelles illimités, Bull. Calcatta Math. Soc. 30 (1938), 9–10.
• H. Shapiro, Composition operators and Schröder's functional equation, Contemp. Math. 213 (1998), 213–228.
• M. P. Wiernsberger, Sur les polygones et les radicaux carrés superposés, Journal Reine Angew. Math. 130 (1905), 144–152.