On simultaneous Abel equations

Summary

We investigate the continuous and measurable solutions of the system of Abel's functional equations

$$\begin{gathered} \varphi (f(x)) = \varphi (x) + 1 \hfill \\ ,x \in (a,b) \hfill \\ \varphi (g(x)) = \varphi (x) + s \hfill \\ \end{gathered} $$

((1))

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 }.