Invariant measures for maps of the interval that do not have points of some period

1994 ◽  
Vol 14 (1) ◽  
pp. 9-21 ◽  
Author(s):  
Jozef Bobok ◽  
Milan Kuchta
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Invariant Measures ◽  
Periodic Points ◽  
Real Interval

AbstractWe study invariant measures for a continuous function which maps a real interval into itself. We show that the ratio of the measures of the two subintervals into which it is divided by a fixed point is constrained by the the set of periods of periodic points. As a consequence of this we give a new forcing relation between periodic points.

scholarly journals Generalizations of the Poincaré Birkhoff fixed point theorem

1977 ◽  
Vol 17 (3) ◽  
pp. 375-389 ◽  
Author(s):  
Walter D. Neumann
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Periodic Points ◽  
Open Problems ◽  
Number Of Components ◽  
Birkhoff 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.

scholarly journals Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

2018 ◽  
Vol 28 (04) ◽  
pp. 1830011
Author(s):  
Mio Kobayashi ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Chaotic Behavior ◽  
Periodic Points ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Bifurcation Structure ◽  
Unstable Fixed Point ◽  
Gaussian Map ◽  
Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

scholarly journals Optimal results in Lorentzian Aubry–Mather theory

2020 ◽  
Author(s):  
Stefan Suhr
Keyword(s):  
Fixed Point ◽  
Invariant Measures ◽  
Lipschitz Continuity ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Multiplicity Results ◽  
Time Separation ◽  
Abelian Cover ◽  
Mather Theory ◽  
Maximal Invariant

AbstractThis article complements the Lorentzian Aubry–Mather Theory in Suhr (Geom Dedicata 160:91–117, 2012; J Fixed Point Theory Appl 21:71, 2019) by giving optimal multiplicity results for the number of maximal invariant measures. As an application the optimal Lipschitz continuity of the time separation on the Abelian cover is established.

scholarly journals Non-Unique Fixed Point Theorems in Modular Metric Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 549
Author(s):  
Sharafat Hussain
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Unique Fixed Point ◽  
Periodic Points ◽  
Modular Metric Spaces ◽  
Modular Spaces

This paper is devoted to the study of Ćirić-type non-unique fixed point results in modular metric spaces. We obtain various theorems about a fixed point and periodic points for a self-map on modular spaces which are not necessarily continuous and satisfy certain contractive conditions. Our results extend the results of Ćirić, Pachpatte, and Achari in modular metric spaces.

scholarly journals Existence of Periodic Fixed Point Theorems in the Setting of Generalized Quasi-Metric Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Chi-Ming Chen ◽  
Erdal Karapınar ◽  
Vladimir Rakočević
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Periodic Points ◽  
Contractive Mappings

We introduce the notions of(α-ϕ-ψ)-weaker Meir-Keeler contractive mappings and(α-φ)-stronger Meir-Keeler contractive mappings. We discuss the existence of periodic points in the setting of generalized quasi-metric spaces. Our results improve, extend, and generalize several results in the literature.

scholarly journals Examples of expanding maps with some special properties

1987 ◽  
Vol 36 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Periodic Point ◽  
Topological Entropy ◽  
Real Line ◽  
Periodic Points ◽  
Unit Interval ◽  
Expanding Maps ◽  
The Real ◽  
Piecewise Monotonic

Let I be the unit interval [0, 1] of the real line. For integers k ≥ 1 and n ≥ 2, we construct simple piecewise monotonic expanding maps Fk, n in C0 (I, I) with the following three properties: (1) The positive integer n is an expanding constant for Fk, n for all k; (2) The topological entropy of Fk, n is greater than or equal to log n for all k; (3) Fk, n has periodic points of least period 2k · 3, but no periodic point of least period 2k−1 (2m+1) for any positive integer m. This is in contrast to the fact that there are expanding (but not piecewise monotonic) maps in C0(I, I) with very large expanding constants which have exactly one fixed point, say, at x = 1, but no other periodic point.

One-Signed Harmonic Solutions and Sign-Changing Subharmonic Solutions to Scalar Second Order Differential Equations

2012 ◽  
Vol 12 (3) ◽  
Author(s):  
Alberto Boscaggin
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Point Theorem ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Subharmonic Solutions

AbstractUsing a recent modified version of the Poincaré-Birkhoff fixed point theorem [19], we study the existence of one-signed T-periodic solutions and sign-changing subharmonic solutions to the second order scalar ODEu′′ + f (t, u) = 0,being f : ℝ × ℝ → ℝ a continuous function T-periodic in the first variable and such that f (t, 0) ≡ 0. Partial extensions of the results to a general planar Hamiltonian systems are given, as well.

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