On iteration groups of C1-diffeomorphisms with two fixed points

Abstract Let I = [ 0 , 1 ] and let f : I → I satisfy the following general hypothesis: f ∈ Diff 1 [ I ] , f ( 0 ) = 0 , f ( 1 ) = 1 , f ( x ) ≠ x , x ∈ Int I and the condition (A): f ′ ( x ) = s + 0 ( x δ ) , x → 0 , and f ′ ( x ) = M + 0 ( | x − 1 | δ ) , x → 1 , with s 1 M . If { f t : I → I , t ∈ R } is a continuous iteration group where all functions are of class C 1 and at least one of the iterates satisfies condition (A), then there exists a diffeomorphism ψ : I → I such that f t = ψ − 1 ∘ p k t ∘ ψ , t ∈ R , where p k t ( x ) : = s t x [ 1 + ( s t k − 1 ) x k ] 1 / k , x ∈ I and k = − log M log s . The function ψ is given by the formula ψ ( x ) = lim n → ∞ f n ( x ) [ ( f n ( x 0 ) ) k + ( f n ( x ) ) k ] 1 / k , x ∈ I with an arbitrary fixed x 0 ∈ Int I . Giving an example of a C 1 -iteration group with two fixed points which does not conjugate diffeomorphically with the group { p k t : I → I , t ∈ R } we show that some additional assumption on diffeomorphism f is essential.