ON FUNCTIONS ATTRACTING POSITIVE ENTROPY

2017 ◽  
Vol 97 (1) ◽  
pp. 69-79 ◽  
Author(s):  
ANNA LORANTY ◽  
RYSZARD J. PAWLAK
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Lebesgue Measure ◽  
Positive Entropy ◽  
Density Point

We examine dynamical systems which are ‘nonchaotic’ on a big (in the sense of Lebesgue measure) set in each neighbourhood of a fixed point $x_{0}$, that is, the entropy of this system is zero on a set for which $x_{0}$ is a density point. Considerations connected with this family of functions are linked with functions attracting positive entropy at $x_{0}$, that is, each mapping sufficiently close to the function has positive entropy on each neighbourhood of $x_{0}$.

scholarly journals A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds

1987 ◽  
Vol 7 (3) ◽  
pp. 463-479 ◽  
Author(s):  
Stephan Pelikan ◽  
Edward E. Slaminka
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Lebesgue Measure ◽  
Upper Bound ◽  
Fixed Point Index ◽  
Compact Sets ◽  
Point Index ◽  
Area Preserving Maps ◽  
Area Preserving

AbstractThe study of area preserving maps of manifolds has an extensive history in the theory of dynamical systems. One interest has been in the behaviour of such maps near an isolated fixed point. In 1974 Carl Simon proved the existence of an upper bound for the index of an isolated fixed point for Ck area preserving diffeomorphisms of a surface. We extend his result to homeomorphisms of an orientable two manifold. The proof utilizes the notion of free modification, developed by Morton Brown, and enlarges the scope of the problem to the consideration of ‘nice’ measures, i.e. uniformly equivalent to Lebesgue measure on compact sets. By suitably modifying the homeomorphism and the measure, we obtain the following theorem.

scholarly journals On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1088 ◽  
Author(s):  
Juan A. Aledo ◽  
Ali Barzanouni ◽  
Ghazaleh Malekbala ◽  
Leila Sharifan ◽  
Jose C. Valverde
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Periodic Orbits ◽  
Periodic Structure ◽  
Boolean Functions ◽  
Directed Graphs ◽  
General Pattern ◽  
General Situation ◽  
Point System ◽  
Local Functions

In this paper, based on previous results on AND-OR parallel dynamical systems over directed graphs, we give a more general pattern of local functions that also provides fixed point systems. Moreover, by considering independent sets, this pattern is also generalized to get systems in which periodic orbits are only fixed points or 2-periodic orbits. The results obtained are also applicable to homogeneous systems. On the other hand, we study the periodic structure of parallel dynamical systems given by the composition of two parallel systems, which are conjugate under an invertible map in which the inverse is equal to the original map. This allows us to prove that the composition of any parallel system on a maxterm (or minterm) Boolean function and its conjugate one by means of the complement map is a fixed point system, when the associated graph is undirected. However, when the associated graph is directed, we demonstrate that the corresponding composition may have points of any period, even if we restrict ourselves to the simplest maxterm OR and the simplest minterm AND. In spite of this general situation, we prove that, when the associated digraph is acyclic, the composition of OR and AND is a fixed point system.

scholarly journals Global Dynamical Systems Involving Generalized -Projection Operators and Set-Valued Perturbation in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yun-zhi Zou ◽  
Xi Li ◽  
Nan-jing Huang ◽  
Chang-yin Sun
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Banach Spaces ◽  
Equilibrium Points ◽  
Projection Operators ◽  
Initial Value ◽  
New Class ◽  
The Fixed Point Theorem ◽  
Generalized Projection Operators ◽  
Generalized Dynamical Systems

A new class of generalized dynamical systems involving generalizedf-projection operators is introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the equilibrium points set of this class of generalized global dynamical systems is proved to be nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems with set-valued perturbation is showed to be continuous with respect to the initial value.

scholarly journals On the controllability of fractional dynamical systems

2012 ◽  
Vol 22 (3) ◽  
pp. 523-531 ◽  
Author(s):  
Krishnan Balachandran ◽  
Jayakumar Kokila
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Matrix Function ◽  
Sufficient Conditions ◽  
Schauder’S Fixed Point Theorem ◽  
Finite Dimensional ◽  
Linear And Nonlinear ◽  
Controllability Grammian

Abstract This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder’s fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.

scholarly journals Rigidity theory for C∗-dynamical systems and the “Pedersen rigidity problem”

2018 ◽  
Vol 29 (03) ◽  
pp. 1850016 ◽  
Author(s):  
S. Kaliszewski ◽  
Tron Omland ◽  
John Quigg
Keyword(s):  
Fixed Point ◽  
Abelian Group ◽  
Dynamical Systems ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Alternative Formulation ◽  
Crossed Products ◽  
Locally Compact ◽  
Fixed Point Algebra ◽  
Multiplier Algebras

Let [Formula: see text] be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of [Formula: see text] on [Formula: see text]-algebras [Formula: see text] and [Formula: see text] are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of [Formula: see text] and [Formula: see text] in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of [Formula: see text] and [Formula: see text]. There is an alternative formulation of the problem: an action of the dual group [Formula: see text] together with a suitably equivariant unitary homomorphism of [Formula: see text] give rise to a generalized fixed-point algebra via Landstad’s theorem, and a problem related to the above is to produce an action of [Formula: see text] and two such equivariant unitary homomorphisms of [Formula: see text] that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of [Formula: see text] and [Formula: see text] is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if [Formula: see text] is discrete, this will be the case for all actions of [Formula: see text].

ON LOCAL CONTROL OF CHAOS: THE NEIGHBOURHOOD SIZE

1996 ◽  
Vol 06 (01) ◽  
pp. 169-178 ◽  
Author(s):  
MIRKO PASKOTA
Keyword(s):  
Functional Analysis ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Local Control ◽  
Linear Control ◽  
Control Law ◽  
Control Of Chaos ◽  
Stability Margins ◽  
Chaotic Dynamical Systems ◽  
The Stability

In this paper, conditions for effective local control of nonlinear, chaotic dynamical systems are analyzed. The aim is to determine the size of a neighbourhood of a fixed point in which the system remains stable when subjected to a linear control law. The stability margins are obtained using ideas from functional analysis, and ways of implementing the obtained results in practice are proposed. An implementation of obtained conditions is illustrated on a real application.

Entropy of $C^\ast$-dynamical systems defined by bitstreams

1998 ◽  
Vol 18 (4) ◽  
pp. 859-874 ◽  
Author(s):  
V. YA. GOLODETS ◽  
ERLING ST&\Oslash;RMER
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Positive Entropy ◽  
Completely Positive ◽  
Car Algebra

We study automorphisms of the CAR-algebra obtained from binary shifts. We consider cases when the $C^\ast$-dynamical system is asymptotically abelian, is proximally asymptotically abelian, is an entropic $K$-system or has completely positive entropy. The entropy is computed in several cases.

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