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Journal: Annales Mathematicae Silesianae

On conjugacy of disjoint iteration groups on the unit circle

Krzysztof Ciepliński

Language: EN | Published: 30-09-1999 | Abstract | pp. 103-118

The aim of this paper is to give a necessary and sufficient condition for conjugacy of some iteration groups ???? = {F^t : S↦S, t∈ℝ} and ???? = {G^t : S↦S, t∈ℝ} defined on the unit circle S. Our basic assumption is that the yare non-singular, that is at least one element of ???? and ???? has no periodic point. Moreover, under some further restrictions, we determine all orientation-preserving homeomorphisms Γ : S↦S such that
Γ◦F^t = G^t◦Γ, t∈ℝ.

1999-09-30



Journal: Annales Mathematicae Silesianae

Differentiable solutions of functional equations in Banach spaces

Maciej Sablik

Language: EN | Published: 30-09-1993 | Abstract | pp. 17-55

We deal with the functional equation
ϕ(F(x)) = g(x,ϕ(x))
where functions F and g are given, defined in open subsets of Banach spaces and taking values in Banach spaces as well. We prove theorems on the existtence and uniqueness of solutions of the equation in classes of differentiable functions. As corollaries we get some results on the conjugacy of diffeomorphisms. Analogous results have been known in the finite dimensional case only.

1993-09-30