Bi-iterative limits used to the theory of the Schröder equation

Let f be a fixed point free increasing homeomorphism of R+ onto itself such that the limit d:=limn→∞⁡fn+1(x)fn(x) exits, belongs to the interval (0,1) and is independent of x∈R+. For each α∈R+ we define the α-bi-iterative limits fα,∞_ and fα,∞‾ of f to be the lower and the upper limits of the sequence {f−n(αfn(x)):n∈N,x∈R+}, respectively. We show that the following statements are equivalent: (a) The Schröder equation σ(f(x))=dσ(x) has a continuous regularly varying solution. (b) The set consisting of the differentials at zero of the bi-iterative limits of f is dense in the multiplicative group R+. (c) There exists α∈R+ such that β:=limx→0+⁡fα,∞_(x)/x exists, belongs to R+ and log⁡β/log⁡d is irrational.