General construction of non-dense disjoint iteration groups on the circle

Krzysztof Ciepliński

Displaying similar documents to “General construction of non-dense disjoint iteration groups on the circle”

The structure of disjoint iteration groups on the circle

Krzysztof Ciepliński (2004)

Czechoslovak Mathematical Journal

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The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle $𝕊^{1}$ , that is, families $ℱ = {F^{v} 𝕊^{1} ⟶ 𝕊^{1} v \in V}$ of homeomorphisms such that $F^{v_{1}} \circ F^{v_{2}} = F^{v_{1} + v_{2}}, v_{1}, v_{2} \in V,$ and each $F^{v}$ either is the identity mapping or has no fixed point ( $(V, +)$ is an arbitrary $2$ -divisible nontrivial (i.e., $c a r d V > 1$ ) abelian group).

A dual-parameter double-step splitting iteration method for solving complex symmetric linear equations

Beibei Li, Jingjing Cui, Zhengge Huang, Xiaofeng Xie (2024)

Applications of Mathematics

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We multiply both sides of the complex symmetric linear system $A x = b$ by $1 - i ω$ to obtain a new equivalent linear system, then a dual-parameter double-step splitting (DDSS) method is established for solving the new linear system. In addition, we present an upper bound for the spectral radius of iteration matrix of the DDSS method and obtain its quasi-optimal parameter. Theoretical analyses demonstrate that the new method is convergent when some conditions are satisfied. Some tested examples are...

Fixed point approximation under Mann iteration beyond Ishikawa

Anthony Hester, Claudio H. Morales (2020)

Commentationes Mathematicae Universitatis Carolinae

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Consider the Mann iteration $x_{n + 1} = (1 - α_{n}) x_{n} + α_{n} T x_{n}$ for a nonexpansive mapping $T : K \to K$ defined on some subset $K$ of the normed space $X$ . We present an innovative proof of the Ishikawa almost fixed point principle for nonexpansive mapping that reveals deeper aspects of the behavior of the process. This fact allows us, among other results, to derive convergence of the process under the assumption of existence of an accumulation point of ${x_{n}}$ .

Solving variational inclusions by a multipoint iteration method under center-Hölder continuity conditions

Catherine Cabuzel, Alain Pietrus (2007)

Applicationes Mathematicae

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We prove the existence of a sequence $(x_{k})$ satisfying $0 \in f (x_{k}) + \sum_{i = 1}^{M} a_{i} \nabla f (x_{k} + β_{i} (x_{k + 1} - x_{k})) (x_{k + 1} - x_{k}) + F (x_{k + 1})$ , where f is a function whose second order Fréchet derivative ∇²f satifies a center-Hölder condition and F is a set-valued map from a Banach space X to the subsets of a Banach space Y. We show that the convergence of this method is superquadratic.

Defining complete and observable chaos

Víctor Jiménez López (1996)

Annales Polonici Mathematici

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For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which $l i m i n f_{n \to \infty} | f^{n} (x) - f^{n} (y) | = 0$ and $l i m s u p_{n \to \infty} | f^{n} (x) - f^{n} (y) | > 0$ . We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible...

The new iteration methods for solving absolute value equations

Rashid Ali, Kejia Pan (2023)

Applications of Mathematics

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Many problems in operations research, management science, and engineering fields lead to the solution of absolute value equations. In this study, we propose two new iteration methods for solving absolute value equations $A x - | x | = b$ , where $A \in ℝ^{n \times n}$ is an $M$ -matrix or strictly diagonally dominant matrix, $b \in ℝ^{n}$ and $x \in ℝ^{n}$ is an unknown solution vector. Furthermore, we discuss the convergence of the proposed two methods under suitable assumptions. Numerical experiments are given to verify the feasibility, robustness...

Regular fractional iteration of convex functions

Marek Kuczma (1980)

Annales Polonici Mathematici

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The existence of a unique $C^{1}$ solution φ of equation (1) is proved under the condition that f: I → I is convex or concave and of class $C^{1}$ in I, 0 < f(x) < x in I*, and f’(x) > 0 in I. Here I = [0, a] or [0, a), 0 < a ≤ ∞, and I* = I 0.

On some iteration semigroups

Janusz Brzdęk (1995)

Archivum Mathematicum

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Let $F$ be a disjoint iteration semigroup of $C^{n}$ diffeomorphisms mapping a real open interval $I \neq ⌀$ onto $I$ . It is proved that if $F$ has a dense orbit possesing a subset of the second category with the Baire property, then $F = {f_{t} f_{t} (x) = f^{- 1} (f (x) + t) for every x \in I, t \in R}$ 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].