On conjugacy of some systems of functions

Abstract

We investigate the bounded solutions \( \varphi:[0,1]\to X \) of the system of functional equations¶¶\( \varphi(f_k(x))=F_k(\varphi(x)),\;\;k=0,\ldots,n-1,x\in[0,1] \),(*)¶where X is a complete metric space, \( f_0,\ldots,f_{n-1}:[0,1]\to[0,1] \) and \( F_0,...,F_{n-1}:X\to X \) are continuous functions fulfilling the boundary conditions \( f_{0}(0) = 0, f_{n-1}(1) = 1, f_{k+1}(0) = f_{k}(1), F_{0}(a) = a,F_{n-1}(b) = b,F_{k+1}(a) = F_{k}(b),\,k = 0,\ldots,n-2 \), for some \( a,b\in X \). We give assumptions on the functions f k and F k which imply the existence, uniqueness and continuity of bounded solutions of the system (*). In the case \( X= \Bbb C \) we consider some particular systems (*) of which the solutions determine some peculiar curves generating some fractals. If X is a closed interval we give a collection of conditions which imply respectively the existence of homeomorphic solutions, singular solutions and a.e. nondifferentiable solutions of (*).