Chaotic actions of topological semigroups

Schneider, Friedrich Martin; Kerkhoff, Sebastian; Behrisch, Mike; Siegmund, Stefan

doi:10.1007/s00233-013-9517-4

Cited by 29 publications

(18 citation statements)

References 12 publications

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“…Definition 2.9 in §2.2) ⇒ Sensitivity to initial conditions. This therefore generalizes the available main results in literature such as [9, Theorem 1.2] and [39,Theorem 4.7] which are only for discrete group T . Moreover, because the topology of T is not necessarily to be discrete here, our result is useful for the classical C 0 -semiflows on manifolds with T = (R, +).…”

Section: Sensitivity Of Semiflowssupporting

confidence: 84%

“…It is well known that sensitive dependence on initial conditions is one of the basic ideas in chaotic dynamics; see, for example, [33,7,16,8,24,1,28,3,23,30,42,39,26,47,43,11,12,32] and so on. First of all we need to recall the notion of sensitivity in the general setting.…”

Section: Sensitivity Of Semiflowsmentioning

confidence: 99%

“…In addition, it should be mentioned that our approaches are completely different with those introduced in [8,28,30,9,39].…”

Section: Sensitivity Of Semiflowsmentioning

confidence: 99%

“…On a Hausdorff uniform space X, every periodic orbit in the sense of Schneider et al [39] is a compact minimal set. And so two distinct periodic orbits T x and T y in [39] are far away (i.e. one can find an index β ∈ U X with β[T x] ∩ β[T y] = ∅).…”

Section: Devaney Chaos Of Semiflowsmentioning

confidence: 99%

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Devaney chaos, Li–Yorke chaos, and multi-dimensional Li–Yorke chaos for topological dynamics

Dai¹,

Tang²

2017

Journal of Differential Equations

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Let π : T × X → X, written T π X, be a topological semiflow/flow on a uniform space X with T a multiplicative topological semigroup/group not necessarily discrete. We then prove:• If T π X is non-minimal topologically transitive with dense almost periodic points, then it is sensitive to initial conditions. As a result of this, Devaney chaos ⇒ Sensitivity to initial conditions, for this very general setting.Let R + π X be a C 0 -semiflow on a Polish space; then we show:• If R + π X is topologically transitive with at least one periodic point p and there is a dense orbit with no nonempty interior, then it is multi-dimensional Li-Yorke chaotic; that is, there is a uncountable set Θ ⊆ X such that for any k ≥ 2 and any distinct points x 1 , . . . , x k ∈ Θ, one can find two time sequences s n → ∞, t n → ∞ with s n (x 1 , . . . , x k ) → (x 1 , . . . , x k ) ∈ X k and t n (x 1 , . . . , x k ) → (p, . . . , p) ∈ ∆ X k .Consequently, Devaney chaos ⇒ Multi-dimensional Li-Yorke chaos.Moreover, let X be a non-singleton Polish space; then we prove:• Any weakly-mixing C 0 -semiflow R + π X is densely multi-dimensional Li-Yorke chaotic.• Any minimal weakly-mixing topological flow T π X with T abelian is densely multidimensional Li-Yorke chaotic.• Any weakly-mixing topological flow T π X is densely Li-Yorke chaotic.We in addition construct a completely Li-Yorke chaotic minimal SL(2, R)-acting flow on the compact metric space R ∪ {∞}. Our various chaotic dynamics are sensitive to the choices of the topology of the phase semigroup/group T .

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Section: Sensitivity Of Semiflowssupporting

confidence: 84%

Section: Sensitivity Of Semiflowsmentioning

confidence: 99%

“…In addition, it should be mentioned that our approaches are completely different with those introduced in [8,28,30,9,39].…”

Section: Sensitivity Of Semiflowsmentioning

confidence: 99%

Section: Devaney Chaos Of Semiflowsmentioning

confidence: 99%

See 2 more Smart Citations

Devaney chaos, Li–Yorke chaos, and multi-dimensional Li–Yorke chaos for topological dynamics

Dai¹,

Tang²

2017

Journal of Differential Equations

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“…Such function spaces C (X) as state spaces promise a wide variety of dynamic behaviour, much more than just R n . It goes beyond the scope of this article to study them in more detail, but in [38] and [37], we examine topological dynamical systems on function spaces over topological groups and semigroups more closely. In particular, we study and characterise faithful strongly chaotic continuous actions of locally compact Hausdorff topological groups on such spaces.…”

Section: Proposition 46mentioning

confidence: 99%

Dynamical Systems in Categories

Behrisch

Kerkhoff

Pöschel

et al. 2015

Appl Categor Struct

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In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We discuss that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, strongly σ -compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic F. M. M. Behrisch et al. algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.

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