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Fixed point indices and periodic points of holomorphic mappings  [PDF]
Guang Yuan Zhang
Mathematics , 2005,
Abstract: Let $\Delta ^{n}$ be the ball $|x|<1$ in the complex vector space $\mathbb{C}% ^{n}$, let $f:\Delta ^{n}\to \mathbb{C}^{n}$ be a holomorphic mapping and let $M$ be a positive integer. Assume that the origin $% 0=(0,..., 0)$ is an isolated fixed point of both $f$ and the $M$-th iteration $f^{M}$ of $f$. Then for each factor $m$ of $M,$ the origin is again an isolated fixed point of $f^{m}$ and the fixed point index $\mu_{f^{m}}(0)$ of $f^{m}$ at the origin is well defined, and so is the (local) Dold's index (see [\ref{Do}]) at the origin:% \begin{equation*} P_{M}(f,0)=\sum_{\tau \subset P(M)}(-1)^{#\tau}\mu_{f^{M:\tau}}(0), \end{equation*}% where $P(M)$ is the set of all primes dividing $M,$ the sum extends over all subsets $\tau $ of $P(M)$, $#\tau $is the cardinal number of $\tau $ and $% M:\tau =M(\prod_{p\in \tau}p)^{-1}$. $P_{M}(f,0)$ can be interpreted to be the number of periodic points of period $M$ of $f$ overlapped at the origin: any holomorphic mapping $% f_{1}:\Delta ^{n}\to \mathbb{C}^{n}$ sufficiently close to $f$ has exactly $P_{M}(f,0)$ distinct periodic points of period $M$ near the origin$%, $ provided that all the fixed points of $f_{1}^{M}$ near the origin are simple. Note that $f$ itself has no periodic point of period $M$ near the origin$.$ According to M. Shub and D. Sullivan's work [\ref{SS}], a necessary condition so that $P_{M}(f,0)\neq 0$ is that the linear part of $f$ at the origin has a periodic point of period $M.$ The goal of this paper is to prove that this condition is sufficient as well for holomorphic mappings.
Fixed Point Indices and Invariant Periodic Sets of Holomorphic Systems  [PDF]
Guang Yuan Zhang
Mathematics , 2005,
Abstract: This note presents a method to study center families of periodic orbits of complex holomorphic differential equations near singularities, based on some iteration properties of fixed point indices. As an application of this method, we will prove Needham's theorem in a more general version.
Strong algebraization of fixed point properties  [PDF]
Masato Mimura
Mathematics , 2015,
Abstract: Purely algebraic criteria of fixed point properties under relative fixed point property, inspired from Shalom's one in ICM 2006, are established. No bounded generation is imposed. One application is that Steinberg groups St(n,A) over any finitely generated, unital, commutative, and associative ring A, possibly noncommutative, enjoy the fixed point property with respect to any noncommutative L_p-space, provided that n is at least 4 and that p is in (1,infty).
Errata to: Local contribution to the Lefschetz fixed point formula  [PDF]
Mark Goresky,Robert MacPherson
Mathematics , 2003,
Abstract: This note contains a correction to the paper, ``Local contribution to the Lefschetz fixed point formula'', Inv. Math. 111 (1993), pp. 1-33.
Properties of the Fixed Point Lattice Dirac Operator in the Schwinger Model  [PDF]
Federico Farchioni,Victor Laliena
Physics , 1998, DOI: 10.1103/PhysRevD.58.054501
Abstract: We present a numerical study of the properties of the Fixed Point lattice Dirac operator in the Schwinger model. We verify the theoretical bounds on the spectrum, the existence of exact zero modes with definite chirality, and the Index Theorem. We show by explicit computation that it is possible to find an accurate approximation to the Fixed Point Dirac operator containing only very local couplings.
Fixed Point Properties Related to Multivalued Mappings  [cached]
Hidetoshi Komiya
Fixed Point Theory and Applications , 2010, DOI: 10.1155/2010/581728
Abstract: We discuss fixed point properties of convex subsets of locally convex linear topological spaces. We derive equivalence among fixed point properties concerning several types of multivalued mappings.
Fixed Point Properties Related to Multivalued Mappings  [cached]
Komiya Hidetoshi
Fixed Point Theory and Applications , 2010,
Abstract: We discuss fixed point properties of convex subsets of locally convex linear topological spaces. We derive equivalence among fixed point properties concerning several types of multivalued mappings.
Infinite groups with fixed point properties  [PDF]
G. Arzhantseva,M. R. Bridson,T. Januszkiewicz,I. J. Leary,A. Minasyan,J. Swiatkowski
Mathematics , 2007, DOI: 10.2140/gt.2009.13.1229
Abstract: We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any action of $Q$ on any $X\in \mathcal{X}_{ac}$, there is a global fixed point. Moreover, $Q$ may be chosen to be simple and to have Kazhdan's property (T). We construct a finitely presented infinite group $P$ that admits no non-trivial action by diffeomorphisms on any smooth manifold in $\mathcal{X}_{ac}$. In building $Q$, we exhibit new families of hyperbolic groups: for each $n\geq 1$ and each prime $p$, we construct a non-elementary hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper subset of which generates a finite $p$-group.
On Approximate Coincidence Point Properties and Their Applications to Fixed Point Theory
Wei-Shih Du
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/302830
Abstract: We first establish some existence results concerning approximate coincidence point properties and approximate fixed point properties for various types of nonlinear contractive maps in the setting of cone metric spaces and general metric spaces. From these results, we present some new coincidence point and fixed point theorems which generalize Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and some well-known results in the literature.
A note on properties that imply the fixed point property
S. Dhompongsa,A. Kaewkhao
Abstract and Applied Analysis , 2006,
Abstract: We give relationships between some Banach-space geometric properties that guarantee the weak fixed point property. The results extend some known results of Dalby and Xu.
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