Fixed Points of Automorphisms of Free Pro-p Groups of Rank 2

1995 ◽  
Vol 47 (2) ◽  
pp. 383-404 ◽  
Author(s):  
Wolfgang N. Herfort ◽  
Luis Ribes ◽  
Pavel A. Zalesskii
Keyword(s):  
Fixed Points ◽  
Prime Number ◽  
Finite Order ◽  
Finite Power ◽  
Rank 2

AbstractLet p be a prime number, and let F be a free pro-p group of rank two. Consider an automorphism α of F of finite order m, and let FixF(α) = {x ∈ F | α(x) = x} be the subgroup of F consisting of the elements fixed by α. It is known that if m is prime to p and α = idF, then the rank of FixF(α) is infinite. In this paper we show that if m is a finite power pr of p, the rank of FixF(α) is at most 2. We conjecture that if the rank of F is n and the order of a is a power of α, then rank (FixF(α)) ≤ n.

scholarly journals ON SOME PROJECTION METHODS FOR APPROXIMATING FIXED POINTS OF NONLINEAR EQUATIONS IN BANACH SPACE

1990 ◽  
Vol 21 (4) ◽  
pp. 351-357
Author(s):  
IOANNIS K. ARGYROS
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Linear Operator ◽  
Fixed Points ◽  
Finite Order ◽  
Operator Equation ◽  
Algebraic System ◽  
Projection Methods ◽  
Linear Algebraic System ◽  
Non Linear Operator

We use a Newton-like method to approximate a fixed point of a non- linear operator equation in a Banach space. Our iterates are computed at each step by solving a linear algebraic system of finite order.

Lie Tori and Their Fixed Point Subalgebras

2009 ◽  
Vol 16 (03) ◽  
pp. 381-396 ◽  
Author(s):  
Saeid Azam ◽  
Valiollah Khalili
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lie Algebras ◽  
Finite Order ◽  
Affine Lie Algebras ◽  
Order Automorphism

We study the fixed point subalgebra of a centerless irreducible Lie torus under a certain finite order automorphism. We investigate which axioms of a Lie torus hold for the fixed points and which do not. We relate our study to some recent results about the fixed points of extended affine Lie algebras under a class of finite order automorphisms.

On the Theorems of Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk

1984 ◽  
Vol 27 (2) ◽  
pp. 192-204 ◽  
Author(s):  
H. Steinlein
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Fixed Points ◽  
Prime Number ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Asymptotic Fixed Point ◽  
Asymptotic Fixed Point Theory

AbstractLet p ≥ 3 be a prime number and m a positive integer, and let S be the sphere S(m-1)(p-1)-1. Let f:S→S be a map without fixed points and with fp = idS. We show that there exists an h: S→ℝm with h(x) ≠ h(f(x)) for all x ∈ S. From this we conclude that there exists a closed cover U1,…, U4m of S with Uinf(Ui) = Ø for i = 1,…, 4m. We apply these results to Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk theorems in the framework of the sectional category and to a problem in asymptotic fixed point theory.

Newton's method and a class of meromorphic functions without wandering domains

1993 ◽  
Vol 13 (2) ◽  
pp. 231-247 ◽  
Author(s):  
Walter Bergweiler
Keyword(s):  
Fixed Points ◽  
Finite Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Newton Iteration ◽  
Fatou Set ◽  
Iteration Function ◽  
Wandering Domains

AbstractLetNbe the class of meromorphic functionsfwith the following properties:fhas finitely many poles;f′ has finitely many multiple zeros; the superattracting fixed points offare zeros off′ and vice versa, with finitely many exceptions;fhas finite order. It is proved that iff∈N, thenfdoes not have wandering domains. Moreover, iff∈Nand if ∞ is among the limit functions offnin a cycle of periodic domains, then this cycle contains a singularity off−1. (Herefndenotes thenth iterate off) These results are applied to study Newton's method for entire functionsgof the formwherepandqare polynomials and wherecis a constant. In this case, the Newton iteration functionf(z) =z−g(z)/g′(z) is inN. It follows thatfn(z) converges to zeros ofgfor allzin the Fatou set off, if this is the case for all zeroszofg″. Some of the results can be extended to the relaxed Newton method.

scholarly journals Fixed points of differences of meromorphic functions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zhaojun Wu ◽  
Jia Wu
Keyword(s):  
Meromorphic Function ◽  
Fixed Points ◽  
Finite Order ◽  
Complex Number ◽  
Meromorphic Functions ◽  
Acta Math ◽  
Transcendental Meromorphic Function ◽  
Nonzero Complex Number ◽  
Discrete Analogues

Abstract Let f be a transcendental meromorphic function of finite order and c be a nonzero complex number. Define $\Delta _{c}f=f(z+c)-f(z)$ Δ c f = f ( z + c ) − f ( z ) . The authors investigate the existence on the fixed points of $\Delta _{c}f$ Δ c f . The results obtained in this paper may be viewed as discrete analogues on the existing theorem on the fixed points of $f'$ f ′ . The existing theorem on the fixed points of $\Delta _{c}f$ Δ c f generalizes the relevant results obtained by (Chen in Ann. Pol. Math. 109(2):153–163, 2013; Zhang and Chen in Acta Math. Sin. New Ser. 32(10):1189–1202, 2016; Cui and Yang in Acta Math. Sci. 33B(3):773–780, 2013) et al.

scholarly journals On Properties of meromorphic solutions of difference Painlevé I and II equation

2017 ◽  
Vol 5 (2) ◽  
pp. 37
Author(s):  
Weimin Xue ◽  
Yanmei Teng
Keyword(s):  
Fixed Points ◽  
Finite Order ◽  
Divided Difference ◽  
Meromorphic Solutions

In this paper, we investigate some properties of finite order transcendental meromorphic solutions of difference Painlev \(\)\acute{e}\) I and II equations, and obtain precise estimations of exponents of convergence of poles of difference \(\)\Delta w(z)=w(z+1)-w(z)\) and divided difference \(\)\frac{\Delta w(z)}{w(z)}\), and of fixed points of \(\)w(z+\eta)$ ($\eta\in \mathbb{C}\setminus\{0\}\)).

scholarly journals On Block Idempotents of Modular Group Rings

1966 ◽  
Vol 27 (2) ◽  
pp. 429-433 ◽  
Author(s):  
Masaru Osima
Keyword(s):  
Prime Number ◽  
Finite Order ◽  
Number Field ◽  
Algebraic Number ◽  
Modular Group ◽  
Algebraic Number Field ◽  
Roots Of Unity ◽  
Group Rings ◽  
Irreducible Characters

We consider a group G of finite order g = pag′ where p is a prime number and (p, g′) = 1. Let Ω be the algebraic number field which contains the p-th roots of unity. Let K1, K2,…, Kn be the classes of conjugate elements in G and the first m(≦n) classes be p-regular. There exist n distinct (absolutely) irreducible characters x1, x2,…, xn of G.

scholarly journals Relations between Solutions of Differential Equations and Small Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Wei Liu ◽  
Zong-Xuan Chen
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fixed Points ◽  
Finite Order ◽  
Entire Functions ◽  
Small Functions ◽  
Small Growth ◽  
Derivatives Of ◽  
Transcendental Solution

We investigate relations between solutions, their derivatives of differential equationf(k)+Ak−1f(k−1)+⋯+A1f’+A0f=0, and functions of small growth, whereAj  (j=0,1,…,k−1)are entire functions of finite order. By these relations, we see that every transcendental solution and its derivative of above equation have infinitely many fixed points.

Automorphisms of the moduli space of principal $G$-bundles induced by outer automorphisms of $G$

2018 ◽  
Vol 122 (1) ◽  
pp. 53
Author(s):  
Álvaro Antón Sancho
Keyword(s):  
Fixed Points ◽  
Finite Order ◽  
Lie Group ◽  
Moduli Space ◽  
Outer Automorphism ◽  
Structure Group ◽  
Simple Complex ◽  
Outer Automorphisms ◽  
The Way

In this work we study finite-order automorphisms of the moduli space of principal $G$-bundles coming from outer automorphisms of the structure group when $G$ is a simple complex Lie group. We do this by describing the subvarieties of fixed points for the action of that automorphisms on the moduli space of principal $G$-bundles. In particular, we prove that these fixed points are reductions of structure group to the subgroup of fixed points of the outer automorphism. Moreover, we study the way in which these fixed points fall into the stable or nonstable locus of the moduli.

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