Almost periodic functions and infinite compositions of quadratic polynomials (Q2910120)

The author of this paper presents examples of almost periodic functions using infinite compositions of quadratic polynomials. Let \(f(x)\circ g(x):=f(g(x))\) and \(\mathcal{R}_{n=m}^Nf_n(x):=f_m(x)\circ f_{m+1}(x)\circ\cdots\circ f_{N-1}(x)\circ f_{N}(x) =f_m(f_{m+1}(\cdots f_{N-1}(f_{N}(x))\cdots))\). The infinite composition \(\mathcal{R}_{n=1}^\infty f_n(x)\) is defined to be the limit \(\lim_{N\to\infty}\mathcal{R}_{n=1}^Nf_n(x)\).NEWLINENEWLINEThe author proves the following theorem.NEWLINENEWLINELet \(\{\varepsilon_n\}_{n=1}^\infty\) be a sequence of nonnegative real numbers such that \(\sum_{n=1}^\infty\varepsilon_n\) is convergent and \(\varepsilon_{n+1}\leq4\varepsilon_n\) for any \(n\geq1\). Then the entire function NEWLINE\[NEWLINE \Lambda(x):=\left(\mathcal{R}_{n=1}^\infty\left(x+ \frac{x^2}{4^n+\varepsilon_n}\right)\right)\circ(-x^2) NEWLINE\]NEWLINE has the following property\(:\) For any integer \(N\) other than zero and any real number \(x\), we have NEWLINE\[NEWLINE |\Lambda(x+N\pi)-\Lambda(x)|\leq2\sum_{r=1+\text{ord}_2N}^\infty \frac{\varepsilon_r}{1+\varepsilon_r/4^r}\,. NEWLINE\]NEWLINE The symbol \(\text{ord}_2N\) is defined by \(N=d\,2^{\text{ord}_2N}\), where \(d\) is odd.