Sharkovskii’s Periodic Point Theorem

2017 ◽  
pp. 37-44
Keyword(s):  
Periodic Point ◽  
Point Theorem

The Livšic periodic point theorem for non-abelian cocycles

1999 ◽  
Vol 19 (3) ◽  
pp. 687-701 ◽  
Author(s):  
WILLIAM PARRY
Keyword(s):  
Periodic Point ◽  
Point Theorem ◽  
Hyperbolic Systems ◽  
Conjugacy Classes ◽  
Closed Orbits ◽  
Transitivity Condition ◽  
Extended Group ◽  
Metric Group

For hyperbolic systems and for Hölder cocycles with values in a compact metric group, we extend Livšic's periodic point characterisation of coboundaries. Here we show that two such cocycles are cohomologous when their respective ‘weights’ (of closed orbits) coincide. When it is only assumed that they are conjugate, one of the cocycles must (in general) be modified by an isomorphism (which stabilises conjugacy classes) to obtain cohomology. When the group is Lie and when a transitivity condition is satisfied, conjugacy of weights ensures that the cocycles are cohomologous with respect to a finitely extended group.

scholarly journals A topological version of a theorem of Mather on twist maps

1984 ◽  
Vol 4 (4) ◽  
pp. 585-603 ◽  
Author(s):  
Glen Richard Hall
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Point ◽  
Rotation Number ◽  
Point Theorem ◽  
Twist Map ◽  
Twist Maps ◽  
Recent Theorem ◽  
Topological Version ◽  
Area Preserving

AbstractIn this report we show that a twist map of an annulus with a periodic point of rotation number p/q must have a Birkhoff periodic point of rotation number p/q. We use topological techniques so no assumption of area-preservation or circle intersection property is needed. If the map is area-preserving then this theorem andthe fixed point theorem of Birkhoff imply a recent theorem of Aubry and Mather. We also show that periodic orbits of (significantly) smallest period for a twist map must be Birkhoff.

Solution of Volterra integral inclusion in b-metric spaces via new fixed point theorem

2016 ◽  
Vol 2017 (1) ◽  
pp. 17-30 ◽  
Author(s):  
Muhammad Usman Ali ◽  
◽  
Tayyab Kamran ◽  
Mihai Postolache ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Spaces ◽  
Integral Inclusion

A generalization of Reich’s fixed point theorem for multi-valued mappings

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3295-3305 ◽  
Author(s):  
Antonella Nastasi ◽  
Pasquale Vetro
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Contractive Condition ◽  
Complete Metric Spaces ◽  
New Class ◽  
Banach Contraction ◽  
The Banach Contraction Principle

Motivated by a problem concerning multi-valued mappings posed by Reich [S. Reich, Some fixed point problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 57 (1974) 194-198] and a paper of Jleli and Samet [M. Jleli, B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl. 2014:38 (2014) 1-8], we consider a new class of multi-valued mappings that satisfy a ?-contractive condition in complete metric spaces and prove some fixed point theorems. These results generalize Reich?s and Mizoguchi-Takahashi?s fixed point theorems. Some examples are given to show the usability of the obtained results.

Sign in / Sign up

Export Citation Format

Share Document