Generalizations of the Poincaré Birkhoff fixed point theorem

It is shown how George D. Birkhoff's proof of the Poincaré Birkhoff theorem can be modified using ideas of H. Poincaré to give a rather precise lower bound on the number of components of the set of periodic points of the annulus. Some open problems related to this theorem are discussed.

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Counting Periodic Points in Parallel Graph Dynamical Systems

Complexity ◽

10.1155/2020/9708347 ◽

2020 ◽

Vol 2020 ◽

pp. 1-9

Author(s):

Juan A. Aledo ◽

Ali Barzanouni ◽

Ghazaleh Malekbala ◽

Leila Sharifan ◽

Jose C. Valverde

Keyword(s):

Fixed Point ◽

Dynamical Systems ◽

Lower Bound ◽

Fixed Point Theorem ◽

Fixed Points ◽

Point Theorem ◽

Periodic Points ◽

Point Of View ◽

Dominating Sets ◽

Minimum Number

Let F:0,1n⟶0,1n be a parallel dynamical system over an undirected graph with a Boolean maxterm or minterm function as a global evolution operator. It is well known that every periodic point has at most two periods. Actually, periodic points of different periods cannot coexist, and a fixed point theorem is also known. In addition, an upper bound for the number of periodic points of F has been given. In this paper, we complete the study, solving the minimum number of periodic points’ problem for this kind of dynamical systems which has been usually considered from the point of view of complexity. In order to do this, we use methods based on the notions of minimal dominating sets and maximal independent sets in graphs, respectively. More specifically, we find a lower bound for the number of fixed points and a lower bound for the number of 2-periodic points of F. In addition, we provide a formula that allows us to calculate the exact number of fixed points. Furthermore, we provide some conditions under which these lower bounds are attained, thus generalizing the fixed-point theorem and the 2-period theorem for these systems.

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A Fixed Point Theorem of the Hardy and Rogers Kind Endowed with Multiplicative Cone-C Class Functions

Earthline Journal of Mathematical Sciences ◽

10.34198/ejms.2119.169179 ◽

2019 ◽

pp. 169-179

Author(s):

Clement Boateng Ampadu

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Metric Space ◽

Contraction Mapping ◽

Cone Metric Space ◽

Open Problems ◽

Cone Metric

In this paper we introduce the multiplicative version of cone-C class functions [1], and obtain some contraction mapping theorems of the Hardy and Rogers kind in multiplicative cone metric space endowed with such functions. Further, we propose some open problems that are publishable in nature.

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Amenability and Fan–Glicksberg theorem for set-valued mappings

Carpathian Journal of Mathematics ◽

10.37193/cjm.2018.03.08 ◽

2018 ◽

Vol 34 (3) ◽

pp. 341-346

Author(s):

ANTHONY TO-MING LAU ◽

◽

LIANGJIN YAO ◽

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Extension Theorem ◽

Open Problems ◽

Invariant Means ◽

Set Valued Mappings

In this paper, we begin by discussion of some well known results on the existence of left invariant means in the spaces: LUC(S), AP(S) and W AP(S) with Hahn-Banach extension theorem. We then give a new and precise proof of the well known Fan–Glicksberg fixed point theorem. This is then followed by a discussion on some related open problems.

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Periodic orbits of transversal maps

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100073539 ◽

1995 ◽

Vol 118 (1) ◽

pp. 161-181 ◽

Cited By ~ 7

Author(s):

J. Casasayas ◽

J. Llibre ◽

A. Nunes

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Zeta Function ◽

Point Theorem ◽

Compact Manifold ◽

Sufficient Conditions ◽

Periodic Points ◽

Homology Groups ◽

Lefschetz Numbers ◽

Lefschetz Fixed Point Theorem

One of the most useful theorems for proving the existence of fixed points, or more generally, periodic points of a continuous self-map f of a compact manifold, is the Lefschetz fixed point theorem. When studying the periodic points of f it is convenient to use the Lefschetz zeta function Zf(t) of f, which is a generating function for the Lefschetz numbers of all iterates of f. The function Zf(t) is rational in t and can be computed from the homological invariants of f. See Section 2 for a precise definition. Thus there exists a relation, based on the Lefschetz fixed point theorem, between the periodic points of a self-map of a manifold f:M → M and the properties of the induced homomorphism f*i on the homology groups of M. This relation has been used in several papers, namely [F1], [F2], [F3] and [M]. In these papers, sufficient conditions are given for the existence of infinitely many periodic points in the case when all the zeros and poles of the associated Lefschetz zeta function are roots of unit. Here we restrict ourselves to maps defined on manifolds with a certain homology type. For transversal maps f defined on this class of manifolds, it is possible to extend the techniques introduced in [F1], [F3] and [M] in order to obtain information on the set of periods of f. We recover the above mentioned results of J. Franks and T. Matsuoka, and derive new results on the set of periods of f when the associated Lefschetz zeta function has zeros or poles outside the unit circle.

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