Marek Cezary Zdun

On simultaneous Abel equations.

SummaryWe investigate the continuous and measurable solutions of the system of Abels functional equations(1)

Regular fractional iterations

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

On a System of Schröder Equations on the Circle

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. Moreoverfn(x) ≠ gm(x) for everyx∈(a, b) and everyn, m ∈ ℤ such that |n| + |m| ≠ 0.LetL be the set of limit points of {fnº gm(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 {ftt} such that ft1 =f and g ∈ {ft} 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 andfn =gm 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 {ft} such that f1 =f and g ∈ {ft}.

On a limit formula for embeddings of diffeomorphisms in regular iteration semigroups

SummaryWe give a general construction of iteration groups of continuous functionsG:= {ft,t ∈ ℝ}} on an interval (a, b) such thatft(x) ≠ x forx ∈ (a, b) providedft≠ id. Two cases may occur(i)For everys, t ∈ ℝ such thatfs≠ id andft≠ id there existn, m ∈ ℤ such that|n| + |m| ≠ 0 andfns =fmt(ii)There exists, t ∈ ℝ such thatfs≠ id andft≠ id and for everyn, m ∈ ℤ, |n| + |m| ≠ 0, fns≠ fmt.

ON CONTINUOUS ITERATION SEMIGROUPS ON THE CIRCLE

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On Semi-Conjugacy Equation for Homeomorphisms of the Circle

We investigate the existence and uniqueness of solutions of the functional equation , , where are closed intervals, and , are some continuous piecewise monotone functions. A fixed point principle plays a crucial role in the proof of our main result.

On some invariants of conjugacy of disjoint iteration groups

Let M be an arbitrary nonempty set and for t ∈ M be continuous mappings of the unit circle . The aim of this paper is to investigate the existence of solutions (Φ, c), where is a continuous function and , of the following system of Schroder equations The particular case when card M = 1 is also considered.

On approximative embeddability of diffeomorphisms in C1-flows

Let be an open set, be a bijection of class and be a globally attractive fixed point of F. Assume that has a real logarithm and the eigenvalues of F satisfy and . Then for every real matrices A and T such that and , where is a fixed integer, there exists the unique iteration semigroup such that and are of class and and the unique iterative root G of order k of F of class such that . They are given by the formulae The iterative roots and iteration semigroups of class are also considered.