Some functional equations which generate both crinkly broken lines and curves (Q1100308)

A sequence of complex numbers \(a=\{a_ n\}^{\infty}_{n=0}\in {\mathbb{C}}^{{\mathbb{N}}}\) generates a broken line L(a) in the complex plane whose turning points \(\{z_ n\}^{\infty}_{n=0}\) are given by \(z_ 0=0\), \(z_ n=\sum^{n-1}_{k=0}a_ k\) \((n=1,2,...)\). For \((\gamma_ 0,\gamma_ 1,...,\gamma_{n-1})\in {\mathbb{C}}\) \(n\setminus \{0\}\), let \[ T_{\gamma}(a_ 0,a_ 1,...)=(\gamma_ 0a_ 0,\gamma_ 1a_ 0,...,\gamma_{n-1}a_ 0,\gamma_ 0a_ 1,..\quad.,\gamma_{n-1}a_ 1,\gamma_ 0,a_ 2,...), \] \(T_{\gamma}\) replaces a segment \(a_ j\) by segments \(\gamma_ 0a_ j,\gamma_ 1a_ j,...,\gamma_{n-1}a_ j\). Many irregular curves can be generated by taking limits, in the sense of Hausdorff metric, of \(L(T\) \(n_{\gamma}(a))\) (\(\gamma\) is the so- called generator of the resulting fractal curve). The author first investigates broken lines which are invariant under \(T_{\gamma}\). If \({\mathbb{C}}^{{\mathbb{N}}}\) is identified with the ring of formal power series \({\mathbb{C}}[[ z]]\), the condition of invariance leads to the functional equation (in \({\mathbb{C}}[[ z]])\) \[ (*)\quad \psi_{\gamma}(z)f(z\quad n)=f(z), \] where \(\psi_{\gamma}=1+\gamma_ 1z+...+\gamma_{n-1}z^{n-1}\). The solution of (*) is obtained, and it is shown that the (a priori formal) power series has radius of convergence 1; in fact, the natural boundary of the corresponding analytic function is the unit circle (with a few exceptions). Since the unit circle is also the Julia set of \(z\mapsto z\) n, the author is led to consider the more general functional equation \[ (**)\quad \psi (z)f(p(z))=f(z), \] where \(\psi\) is entire and p is a polynomial satisfying certain conditions. He shows that if the immediate stable set \(A_ 0\) of 0 (under \(z\mapsto p(z))\) contains a zero of \(\psi\), then the solution of (**) restricted to \(A_ 0\) has the boundary of \(A_ 0\) as its natural boundary.