We consider the functional equation $f\left(xf\right(x\left)\right)=\varphi \left(f\right(x\left)\right)$ where $\varphi J\to J$ is a given increasing homeomorphism of an open interval $J\subset (0,\infty )$ and $f(0,\infty )\to J$ is an unknown continuous function. In a previous paper we proved that no continuous solution can cross the line $y=p$ where $p$ is a fixed point of $\varphi$, with a possible exception for $p=1$. The range of any non-constant continuous solution is an interval whose end-points are fixed by $\varphi$ and which contains in its interior no fixed point except for $1$. We also gave a characterization of the class of continuous...

Defining the complexity of a green pattern exhibited by an interval map, we give the best bounds of the topological entropy of a pattern with a given complexity. Moreover, we show that the topological entropy attains its strict minimum on the set of patterns with fixed eccentricity m/n at a unimodal X-minimal case. Using a different method, the last result was independently proved in[11].

We consider the Abel equation φ[f(x)] = φ(x) + a on the plane ℝ², where f is a free mapping (i.e. f is an orientation preserving homeomorphism of the plane onto itself with no fixed points). We find all its homeomorphic and diffeomorphic solutions φ having positive Jacobian. Moreover, we give some conditions which are equivalent to f being conjugate to a translation.

 Let $F_i=1,...,N$ be affine mappings of ${ℝ}^n$. It is well known that if there exists j ≤ 1 such that for every ${\sigma }_1,...,{\sigma }_j\in 1,...,N$ the composition (1) $F_{\sigma 1}\circ ...\circ F_{{\sigma }_j}$ is a contraction, then for any infinite sequence ${\sigma }_1,{\sigma }_2,...\in 1,...,N$ and any $z\in {ℝ}^n$, the sequence (2)$F_{\sigma 1}\circ ...\circ F_{{\sigma }_n}\left(z\right)$ is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any $z\in {ℝ}^n$ and any $\sigma ={\sigma }_1,{\sigma }_2,...$ belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every $\sigma ={\sigma }_1,{\sigma }_2,...\in \Sigma$ the composition (1) is a contraction. This result...

We prove that if f: → is Darboux and has a point of prime period different from $2^i$, i = 0,1,..., then the entropy of f is positive. On the other hand, for every set A ⊂ ℕ with 1 ∈ A there is an almost continuous (in the sense of Stallings) function f: → with positive entropy for which the set Per(f) of prime periods of all periodic points is equal to A.

We collect and generalize various known definitions of principal iteration semigroups in the case of multiplier zero and establish connections among them. The common characteristic property of each definition is conjugating of an iteration semigroup to different normal forms. The conjugating functions are expressed by suitable formulas and satisfy either Böttcher’s or Schröder’s functional equation.

We consider iteration of arithmetic and power means and discuss methods for determining their limit. These means appear naturally in connection with some problems in homogenization theory.

Let $F$ be a disjoint iteration semigroup of $C^n$ diffeomorphisms mapping a real open interval $I\ne ⌀$ onto $I$. It is proved that if $F$ has a dense orbit possesing a subset of the second category with the Baire property, then $F=\{f_t{f}_t\left(x\right)=f^{-1}(f\left(x\right)+t)\text{for}\text{every}x\in I,t\in R\}$ for some $C^n$ diffeomorphism $f$ of $I$ onto the set of all reals $R$. The paper generalizes some results of J.A.Baker and G.Blanton [3].

In recent papers the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Here the domain is ℝ. A very general positive linear integral type operator is introduced through a convolution-like iteration of another general positive linear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates, shape preserving and...