On simultaneous Abel equations

Abstract

We investigate the continuous and measurable solutions of the system of Abel's functional equations (1) $$\begin{gathered} \varphi (f(x)) = \varphi (x) + 1 \hfill \\ ,x \in (a,b) \hfill \\ \varphi (g(x)) = \varphi (x) + s \hfill \\ \end{gathered} $$ and their applications to the iteration theory. Let us assume the following hypothesis: (H)f, g: (a, b) → (a, b) are continuous bijections andf o g = g o f. Moreoverf n (x) ≠ g m (x) for everyx∈(a, b) and everyn, m ∈ ℤ such that |n| + |m| ≠ 0. LetL be the set of limit points of {f n o g m (x): n, m ∈ ℤ}, wherex ∈(a, b) (L does not depend ofx). The setL is either a perfect and nowhere dense set orL = 〈a, b〉. Theorem.If f and g satisfy hypothesis (H), then there is a unique s ∈ ℝ such that the system (1) has a continuous solution. For this s the system (1) has a continuous solution ϕ unique up to an additive constant. This solution ϕ is monotonic, ϕ[L ⋂ (a, b)] = ℝ and s is irrational. Moreover ϕ is invertible if and only if L = 〈a, b〉. Corollary.Let f and g satisfy hypothesis (H). Then there exists a continuous iteration group {f t t} such that ft 1 =f and g ∈ {f t } if and only if L = 〈a, b〉. Moreover this iteration group is unique. Further let us assume the following hypothesis: (C)f, g: (a, b) →(a, b) are continuous bijections such thatf(x) ≠ x, g(x) ≠ x forx ∈(a, b), fog=gof andf n =g m for somen, m ∈ ℤ\{0}. Theorem.Let hypothesis (C) be fulfilled. Then the system (1) for s = n/m has a continuous and invertible solution. This solution depends on an arbitrary function. Corollary.Let f and g satisfy hypothesis (C), then there exist infinitely many continuous iteration groups {f t } such that f 1 =f and g ∈ {f t }.