scholarly journals Connecting ergodicity and dimension in dynamical systems

1990 ◽  
Vol 10 (3) ◽  
pp. 451-462 ◽  
Author(s):  
C. D. Cutler
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Probability Measures ◽  
Pointwise Dimension ◽  
Continuous Maps ◽  
Borel Probability ◽  
The Relationship

AbstractIn this paper we make precise the relationship between local or pointwise dimension and the dimension structure of Borel probability measures on metric spaces. Sufficient conditions for exact-dimensionality of the stationary ergodic distributions associated with a dynamical system are obtained. A counterexample is provided to show that ergodicity alone is not sufficient to guarantee exactdimensionality even in the case of continuous maps or flows.

On disjointness of dynamical systems

1979 ◽  
Vol 85 (3) ◽  
pp. 477-491 ◽  
Author(s):  
J. Auslander ◽  
Y. N. Dowker
Keyword(s):  
Dynamical System ◽  
Metric Spaces ◽  
Probability Measures ◽  
Compact Metric Spaces ◽  
Probability Spaces ◽  
Dynamical Properties ◽  
Borel Probability ◽  
The Given ◽  
Special Case ◽  
Measure Preserving Transformations

By a dynamical system we mean one of several related objects: measure preserving transformations on probability spaces (processes), self homeomorphisms of compact metric spaces (compact systems), or a combination of these, namely compact systems provided with invariant Borel probability measures. It is the latter, which we call compact processes, which will be of most interest in this paper. In particular, we will study the dynamical properties of the product of two processes with respect to compatible measures – those measures which project to the given measures on the component spaces. This leads to the notion of disjointness of two processes – the only compatible measure is the product measure. As an application we obtain a theorem, a special case of which gives rise to a class of transformations which preserve normal sequences. Finally, we study a topological analog (topological disjointness) and briefly consider the relation between the two notions of disjointness.

scholarly journals Surjunctivity and topological rigidity of algebraic dynamical systems

2017 ◽  
Vol 39 (3) ◽  
pp. 604-619 ◽  
Author(s):  
SIDDHARTHA BHATTACHARYA ◽  
TULLIO CECCHERINI-SILBERSTEIN ◽  
MICHEL COORNAERT
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Countable Group ◽  
Continuous Group ◽  
Continuous Map ◽  
Algebraic Dynamical Systems ◽  
Metrizable Group ◽  
Group Automorphisms

Let$X$be a compact metrizable group and let$\unicode[STIX]{x1D6E4}$be a countable group acting on$X$by continuous group automorphisms. We give sufficient conditions under which the dynamical system$(X,\unicode[STIX]{x1D6E4})$is surjunctive, i.e. every injective continuous map$\unicode[STIX]{x1D70F}:X\rightarrow X$commuting with the action of$\unicode[STIX]{x1D6E4}$is surjective.

Sufficient conditions for expansive group action

2019 ◽  
Vol 20 (03) ◽  
pp. 2050022 ◽  
Author(s):  
Ali Barzanouni
Keyword(s):  
Solvable Group ◽  
Group Action ◽  
Uniform Space ◽  
Sufficient Conditions ◽  
Open Cover ◽  
Probability Measures ◽  
Continuous Action ◽  
Paracompact Space ◽  
Finite Set ◽  
Borel Probability

Existence of expansivity for group action [Formula: see text] depends on algebraic properties of [Formula: see text] and the topology of [Formula: see text]. We give an expansive action of a solvable group on [Formula: see text] while there is no expansive action of a solvable group on a dendrite [Formula: see text]. We prove that a continuous action [Formula: see text] on a compact metric space [Formula: see text] is expansive if and only if there exists an open cover [Formula: see text] such that for any other open cover [Formula: see text], [Formula: see text] for some finite set [Formula: see text]. In this paper, we introduce the notion of topological expansivity of a group action [Formula: see text] on a [Formula: see text]-paracompact space [Formula: see text]. If a [Formula: see text]-paracompact space [Formula: see text] admits topologically expansive action, then [Formula: see text] is Hausdorff space. We also show that a continuous action [Formula: see text] of a finitely generated group [Formula: see text] on a compact Hausdorff uniform space [Formula: see text] is expansive with an expansive neighborhood [Formula: see text] if and only if for every [Formula: see text] there is an entourage [Formula: see text] such that for every two [Formula: see text]-pseudo orbit [Formula: see text] if [Formula: see text] for all [Formula: see text], then [Formula: see text] for all [Formula: see text]. Finally, we introduce measure [Formula: see text]-expansive actions on a uniform space. The set of all [Formula: see text]-expansive measures with common expansive neighborhood for a group action [Formula: see text] is a convex, closed and [Formula: see text]-invariant subset of the set of all Borel probability measures on [Formula: see text]. Also, we show that a group action [Formula: see text] is expansive if all Borel probability measures are [Formula: see text]-expansive or all Dirac measures [Formula: see text], [Formula: see text], have a common expansive neighborhood.

scholarly journals AN INDICATOR FUNCTION LIMIT THEOREM IN DYNAMICAL SYSTEMS

2011 ◽  
Vol 11 (04) ◽  
pp. 681-690
Author(s):  
OLIVIER DURIEU ◽  
DALIBOR VOLNÝ
Keyword(s):  
Dynamical System ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Indicator Function ◽  
Constructive Proof ◽  
Probability Measures

We give a constructive proof of the following result: in all aperiodic dynamical system, for all sequences (an)n∈ℕ ⊂ ℝ+ such that an ↗ ∞ and [Formula: see text] as n → ∞, there exists a set [Formula: see text] having the property that the sequence of the distributions of [Formula: see text] is dense in the space of all probability measures on ℝ. This extends a result of O. Durieu and D. Volný, Ergod. Th. Dynam. Syst. to the non-ergodic case.

scholarly journals On sums of indicator functions in dynamical systems

2009 ◽  
Vol 30 (5) ◽  
pp. 1419-1430 ◽  
Author(s):  
OLIVIER DURIEU ◽  
DALIBOR VOLNÝ
Keyword(s):  
Dynamical System ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Partial Sums ◽  
Probability Measures ◽  
Ergodic Dynamical System ◽  
Indicator Functions

AbstractIn this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every invertible ergodic dynamical system, for every increasing sequence (an)n∈ℕ⊂ℝ+ such that an↗∞ and an/n→0 as n→∞, there exists a dense Gδ of measurable sets A such that the sequence of the distributions of the partial sums $(\sfrac {1}{a_n})\sum _{i=0}^{n-1}(\ind _A-\mu (A))\circ T^i$ is dense in the set of the probability measures on ℝ.

scholarly journals A Note on Variational Principle of Subsets for Nonautonomous Dynamical Systems

2021 ◽  
pp. 1-16
Author(s):  
Jiao Yang
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Variational Principles ◽  
Borel Probability Measure ◽  
Jel Classification ◽  
Probability Measures ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Case ◽  
Borel Probability ◽  
Discrete Type

Abstract In this paper, we introduce measure-theoretic for Borel probability measures to characterize upper and lower Katok measure-theoretic entropies in discrete type and the measure-theoretic entropy for arbitrary Borel probability measure in nonautonomous case. Then we establish new variational principles for Bowen topological entropy for nonautonomous dynamical systems. JEL classification numbers: 37A35. Keywords: Nonautonomous, Measure-theoretical entropies, Variational principles.

On Weak Asymptotic Periodicity

2020 ◽  
Vol 30 (02) ◽  
pp. 2050030
Author(s):  
Karol Gryszka
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Property ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Conjugacy ◽  
Asymptotic Periodicity ◽  
Necessary And Sufficient

We introduce the asymptotic property associated with recurrence-like behavior of orbits in dynamical systems in general metric spaces. We define a notion of weak asymptotic periodicity and determine its elementary properties and relations including the invariance by topological conjugacy. We use the equicontinuity and the topology of the space to describe necessary and sufficient conditions for the existence of such a behavior.

scholarly journals Dynamics of Weakly Mixing Nonautonomous Systems

2019 ◽  
Vol 29 (09) ◽  
pp. 1950123 ◽  
Author(s):  
Mohammad Salman ◽  
Ruchi Das
Keyword(s):  
Dynamical System ◽  
Sufficient Conditions ◽  
Unit Interval ◽  
Nonautonomous System ◽  
Probability Measures ◽  
Topological Transitivity ◽  
Weak Mixing ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical System ◽  
Weakly Mixing

For a commutative nonautonomous dynamical system we show that topological transitivity of the nonautonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in nonautonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic nonautonomous system on intervals, it is proved that weak mixing implies Devaney chaos. Given a periodic nonautonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.

scholarly journals Consensus Analysis for Third-Order Multiagent Systems in Directed Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Yanfen Cao ◽  
Yuangong Sun
Keyword(s):  
Dynamical Systems ◽  
Multiagent Systems ◽  
Sufficient Conditions ◽  
Laplacian Matrix ◽  
Consensus Problem ◽  
Third Order ◽  
Consensus Analysis ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Theoretical Results

We investigate consensus problem for third-order multiagent dynamical systems in directed graph. Necessary and sufficient conditions to consensus of third-order multiagent systems have been established under three different protocols. Compared with existing results, we focus on the relationship between the scaling strengths and the eigenvalues of the involved Laplacian matrix, which guarantees consensus of third-order multiagent systems. Finally, some simulation examples are given to illustrate the theoretical results.

scholarly journals Kosambi–Cartan–Chern (KCC) theory for higher-order dynamical systems

2016 ◽  
Vol 13 (02) ◽  
pp. 1650014 ◽  
Author(s):  
Tiberiu Harko ◽  
Praiboon Pantaragphong ◽  
Sorin V. Sabau
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Cold Dark Matter ◽  
Evolution Equations ◽  
Dynamical Evolution ◽  
Finsler Spaces ◽  
Stability And Instability ◽  
The Relationship ◽  
Autonomous Dynamical Systems

The Kosambi–Cartan–Chern (KCC) theory represents a powerful mathematical method for the investigation of the properties of dynamical systems. The KCC theory introduces a geometric description of the time evolution of a dynamical system, with the solution curves of the dynamical system described by methods inspired by the theory of geodesics in a Finsler spaces. The evolution of a dynamical system is geometrized by introducing a nonlinear connection, which allows the construction of the KCC covariant derivative, and of the deviation curvature tensor. In the KCC theory, the properties of any dynamical system are described in terms of five geometrical invariants, with the second one giving the Jacobi stability of the system. Usually, the KCC theory is formulated by reducing the dynamical evolution equations to a set of second-order differential equations. In this paper, we introduce and develop the KCC approach for dynamical systems described by systems of arbitrary [Formula: see text]-dimensional first-order differential equations. We investigate in detail the properties of the [Formula: see text]-dimensional autonomous dynamical systems, as well as the relationship between the linear stability and the Jacobi stability. As a main result we find that only even-dimensional dynamical systems can exhibit both Jacobi stability and instability behaviors, while odd-dimensional dynamical systems are always Jacobi unstable, no matter their Lyapunov stability. As applications of the developed formalism we consider the geometrization and the study of the Jacobi stability of the complex dynamical networks, and of the [Formula: see text]-Cold Dark Matter ([Formula: see text]CDM) cosmological models, respectively.

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