Singular Solutions of the Generalized Dhombres Functional Equation

We consider singular solutions of the functional equation f(xf(x)) =phi(f(x)) where phi is a given and f an unknown continuous map R+ -> R+. A solution f is regular if the sets R-f boolean AND (0,1]and R-f boolean AND [1,infinity), where R-f is the range of f, are phi-invariant; otherwise f is singular. We show that for singular solutions the associated dynamical system (R-f , phi vertical bar R-f) can have strange properties unknown for the regular solutions. In particular, we show that phi vertical bar R-f can have a periodic point of period 3 and hence can be chaotic in a strong sense. We also provide an effective method of construction of singular solutions.