scholarly journals Dynamics of multivalued linear operators

2017 ◽  
Vol 15 (1) ◽  
pp. 948-958 ◽  
Author(s):  
Chung-Chuan Chen ◽  
J. Alberto Conejero ◽  
Marko Kostić ◽  
Marina Murillo-Arcila
Keyword(s):  
Linear Operators ◽  
Fréchet Spaces ◽  
Topological Transitivity ◽  
Linear Dynamics ◽  
Devaney Chaos ◽  
Topologically Mixing ◽  
Mixing Property

Abstract We introduce several notions of linear dynamics for multivalued linear operators (MLO’s) between separable Fréchet spaces, such as hypercyclicity, topological transitivity, topologically mixing property, and Devaney chaos. We also consider the case of disjointness, in which any of these properties are simultaneously satisfied by several operators. We revisit some sufficient well-known computable criteria for determining those properties. The analysis of the dynamics of extensions of linear operators to MLO’s is also considered.

Li–Yorke chaos in linear dynamics

2014 ◽  
Vol 35 (6) ◽  
pp. 1723-1745 ◽  
Author(s):  
N. C. BERNARDES ◽  
A. BONILLA ◽  
V. MÜLLER ◽  
A. PERIS
Keyword(s):  
Composition Operators ◽  
Linear Manifold ◽  
Holomorphic Functions ◽  
Linear Operators ◽  
Fréchet Spaces ◽  
Linear Dynamics ◽  
Spaces Of Holomorphic Functions ◽  
Dense Set

We obtain new characterizations of Li–Yorke chaos for linear operators on Banach and Fréchet spaces. We also offer conditions under which an operator admits a dense set or linear manifold of irregular vectors. Some of our general results are applied to composition operators and adjoint multipliers on spaces of holomorphic functions.

scholarly journals Devaney Chaos and Distributional Chaos in the Solution of Certain Partial Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Xavier Barrachina ◽  
J. Alberto Conejero
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Semigroups Of Operators ◽  
Partial Differential ◽  
Linear Dynamics ◽  
Distributional Chaos ◽  
Devaney Chaos

The notion of distributional chaos has been recently added to the study of the linear dynamics of operators andC0-semigroups of operators. We will study this notion of chaos for some examples ofC0-semigroups that are already known to be Devaney chaotic.

Automatic Continuity for Linear Functions Intertwining Continuous Linear Operators on Frechet Spaces

1978 ◽  
Vol 30 (03) ◽  
pp. 518-530 ◽  
Author(s):  
Marc P. Thomas
Keyword(s):  
Banach Spaces ◽  
Functional Calculus ◽  
Linear Operators ◽  
Linear Functions ◽  
General Situation ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Automatic Continuity ◽  
Continuity Theory ◽  
Continuous Linear Operators

Many results concerning the automatic continuity of linear functions intertwining continuous linear operators on Banach spaces have been obtained, chiefly by B. E. Johnson and A. M. Sinclair [1; 2; 3; 5]. The purpose of this paper is essentially to extend this automatic continuity theory to the situation of Fréchet spaces. Our motive is partly to be able to handle the more general situation, since for example, questions about Fréchet spaces and LF spaces arise in connection with the functional calculus.

Chaotic Dynamics of Composition Operators on the Space of Continuous Functions

2017 ◽  
Vol 27 (06) ◽  
pp. 1750084 ◽  
Author(s):  
Zongbin Yin
Keyword(s):  
Fixed Points ◽  
Composition Operator ◽  
Chaotic Dynamics ◽  
Composition Operators ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Distributional Chaos ◽  
Long Time ◽  
Topologically Mixing ◽  
Mixing Property

In this paper, the chaotic dynamics of composition operators on the space of real-valued continuous functions is investigated. It is proved that the hypercyclicity, topologically mixing property, Devaney chaos, frequent hypercyclicity and the specification property of the composition operator are equivalent to each other and are stronger than dense distributional chaos. Moreover, the composition operator [Formula: see text] exhibits dense Li–Yorke chaos if and only if it is densely distributionally chaotic, if and only if the symbol [Formula: see text] admits no fixed points. Finally, the long-time behaviors of the composition operator with affine symbol are classified in detail.

Accessibility on iterated function systems

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Maliheh Mohtashamipour ◽  
Alireza Zamani Bahabadi
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Sufficient Condition ◽  
Skew Product ◽  
Iterated Function ◽  
Topologically Mixing ◽  
Mixing Property

AbstractIn this paper, we define accessibility on an iterated function system (IFS) and show that it provides a sufficient condition for the transitivity of this system and its corresponding skew product. Then, by means of a certain tool, we obtain the topologically mixing property. We also give some results about the ergodicity and stability of accessibility and, further, illustrate accessibility by some examples.

scholarly journals Transient Dynamics of Nonsymmetric Perturbations versus Symmetric Instability in Baroclinic Zonal Shear Flows

2010 ◽  
Vol 67 (9) ◽  
pp. 2972-2989 ◽  
Author(s):  
G. R. Mamatsashvili ◽  
V. S. Avsarkisov ◽  
G. D. Chagelishvili ◽  
R. G. Chanishvili ◽  
M. V. Kalashnik
Keyword(s):  
Shear Flows ◽  
Stable Stratification ◽  
Linear Operators ◽  
Vertical Shear ◽  
Transient Dynamics ◽  
Transient Growth ◽  
Coriolis Parameter ◽  
Linear Dynamics ◽  
Wide Range ◽  
Symmetric Instability

Abstract The linear dynamics of symmetric and nonsymmetric perturbations in unbounded zonal inviscid flows with a constant vertical shear of velocity, when a fluid is incompressible and density is stably stratified along the vertical and meridional directions, is investigated. A small–Richardson number Ri ≲ 1 and large–Rossby number Ro ≳ 1 regime is considered, which satisfies the condition for symmetric instability. Specific features of this dynamics are closely related to the nonnormality of linear operators in shear flows and are well interpreted in the framework of the nonmodal approach by tracing the linear dynamics of spatial Fourier harmonics (Kelvin modes) of perturbations in time. The roles of stable stratification, the Coriolis parameter, and vertical shear in the dynamics of perturbations are analyzed. Classification of perturbations into two types or modes—vortex (i.e., quasigeostrophic balanced motions) and inertia–gravity wave—is made according to the value of potential vorticity. The emerging picture of the (linear) transient dynamics for these two modes at Ri ≲ 1 and Ro ≳ 1 indicates that vortex mode perturbations are able to gain basic flow energy and undergo exponential transient amplification and in this process generate inertia–gravity waves. Transient growth of the vortex mode and, consequently, the effectiveness of the wave generation both increase with decreasing Ri and increasing Ro. This linear coupling of perturbation modes is, in general, specific to shear flows but is not fully appreciated yet. A parallel analysis of the transient dynamics of nonsymmetric perturbations versus symmetric instability is also presented. It is shown that the nonnormality-induced transient growth of nonsymmetric perturbations can prevail over the symmetric instability for a wide range of Ri and Ro. The current analysis suggests that the dynamical activity of fronts and jet streaks at Ri ≲ 1 and Ro ≳ 1 should be determined by nonsymmetric perturbations rather than by symmetric ones, as was accepted in earlier papers. It is noteworthy that the transient growth of perturbations is asymmetric in the wavenumber space—the constant phase plane of maximally amplified perturbations is inclined in a direction northeast to the zonal one and the inclination angle is different for different Ri and Ro.

A version of a theorem of Pliss for non-uniform and non-invertible dichotomies

2016 ◽  
Vol 147 (2) ◽  
pp. 225-243 ◽  
Author(s):  
Luis Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Continuous Time ◽  
Transversality Condition ◽  
Linear Operators ◽  
Bounded Solutions ◽  
Linear Dynamics ◽  
Exponential Dichotomies ◽  
The Past ◽  
Bounded Perturbation ◽  
Exponential Behaviour

For a dynamics on the whole line, for both discrete and continuous time, we extend a result of Pliss that gives a characterization of the notion of a trichotomy in various directions. More precisely, the result gives a characterization in terms of an admissibility property in the whole line (namely, the existence of bounded solutions of a linear dynamics under any nonlinear bounded perturbation) of the existence of a trichotomy, i.e. of exponential dichotomies in the future and in the past, together with a certain transversality condition at time zero. In particular, we consider arbitrary linear operators acting on a Banach space as well as sequences of norms instead of a single norm, which allows us to consider the general case of non-uniform exponential behaviour.

Topologically transitive skew-products of operators

2009 ◽  
Vol 30 (1) ◽  
pp. 33-49 ◽  
Author(s):  
FRÉDÉRIC BAYART ◽  
GEORGE COSTAKIS ◽  
DEMETRIS HADJILOUCAS
Keyword(s):  
Linear Operators ◽  
Skew Product ◽  
Weighted Shifts ◽  
Linear Dynamics ◽  
Topologically Transitive ◽  
Skew Products ◽  
Product Systems ◽  
Hypercyclic Vectors ◽  
Backward Shift

AbstractThe purpose of the present paper is to provide a link between skew-product systems and linear dynamics. In particular, we give a criterion for skew-products of linear operators to be topologically transitive. This is then applied to certain families of linear operators including scalar multiples of the backward shift, backward unilateral weighted shifts, composition, translation and differentiation operators. We also prove the existence of common hypercyclic vectors for certain families of skew-product systems.

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