Minimal dynamical systems with closed relations

Banič, Iztok; Erceg, Goran; Rogina, Rene Gril; Kennedy, Judy

doi:10.48550/arxiv.2205.02907

Cited by 3 publications

(10 citation statements)

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“…Thus, for (X, G) we have: Minimal and suitably minimal. Some of our definitions and results overlap with [4] and some results of [4] are presented here in much simpler form.…”

Section: 2mentioning

confidence: 97%

“…We recall basics from [1,2] where dynamical systems are presented by closed relations, aided with some more rigourous discussions in the recent articles [4,5]. Though, our definitions are directed to develop our theory on transitivity for CR-dynamical systems.…”

Section: Cr-dynamical Systemsmentioning

confidence: 99%

“…Let (X, G) be a CR-dynamical system. Then A ⊂ X is called weakly invariant in (X, G) if for every x ∈ A, there exists y ∈ A such that (x, y) ∈ G. (It is called 1-invariant in [4].) Remark 3.6.…”

Section: Cr-dynamical Systemsmentioning

confidence: 99%

“…However, both [1] and [2] overlook any study of the important and one of the oldest dynamical property of 'transitivity' for dynamics given by closed relations. Though, in some way such a study has been made recently in [4,5,6].…”

Section: Introductionmentioning

confidence: 99%

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Revisiting variations in topological transitivity

Nagar

2021

European Journal of Mathematics

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We consider the topological dynamics of closed relations(CR) by studying one of the oldest dynamical property -'transitivity'. We investigate the two kinds of (closed relation) CR-dynamical systems -(X, G) where the relation G ⊆ X × X is closed and (X, G, •) giving the 'suitable dynamics' for a suitable closed relation G, where X is assumed to be a compact metric space without isolated points.(X, G) gives a general approach to study initial value problems for a set of initial conditions, whereas (X, G, •) gives a general approach to study the dynamics of both continuous and quasi-continuous maps.We observe that the dynamics of closed relations is richer than the dynamics of maps and find that we have much more versions of transitivity for these closed relations than what is known for maps.

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“…Thus, for (X, G) we have: Minimal and suitably minimal. Some of our definitions and results overlap with [4] and some results of [4] are presented here in much simpler form.…”

Section: 2mentioning

confidence: 97%

Section: Cr-dynamical Systemsmentioning

confidence: 99%

Section: Cr-dynamical Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Revisiting variations in topological transitivity

Nagar

2021

European Journal of Mathematics

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“…Also, several papers on the topic of dynamical systems with (upper semicontinuous) set-valued functions have appeared recently, see [BEGK,CP,LP,LYY,LWZ,KN,KW,MRT,R,SS,SS2], where more references may be found. However, there is not much known of such dynamical systems and therefore, there are many properties of such set-valued dynamical systems that are yet to be studied.…”

Section: Introductionmentioning

confidence: 99%

Transitive points in CR-dynamical systems

Banič¹,

Erceg²,

Greenwood³

et al. 2022

Preprint

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Towards the Complete Classification of the Two Lines Inverse Limit

Rogina

2023

J Dyn Control Syst

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For positive numbers $$r<1$$ r < 1 and $$\rho >1$$ ρ > 1 , let $$L_{r,\rho }$$ L r , ρ be the union of two line segments in $$[0,1] \times {[}0,1{]}$$ [ 0 , 1 ] × [ 0 , 1 ] , one from (0, 0) to (1, r) and the other one from (0, 0) to $$(\frac{1}{\rho },1)$$ ( 1 ρ , 1 ) . In Banič et al. (2022), it was proven for all such r and $$\rho$$ ρ that, if r and $$\rho$$ ρ never connect, then the Mahavier product of $$L_{r,\rho }$$ L r , ρ ’s is homeomorphic to the Lelek fan. In this paper, we show that for all such r and $$\rho$$ ρ , if r and $$\rho$$ ρ do connect, the Mahavier product of $$L_{r,\rho }$$ L r , ρ ’s is a fan F with top v with some additional properties. More precisely, it is the union of a countable family $$\mathcal {C}$$ C of Cantor fans such that for each $$C_1,C_2\in \mathcal {C}$$ C 1 , C 2 ∈ C , if $$C_1\ne C_2$$ C 1 ≠ C 2 , then $$C_1\cap C_2 = \{v\}$$ C 1 ∩ C 2 = { v } and the set of limit points of the set of end-points of F, forms in each arc from v to an end-point, a harmonic sequence. This solves the open problem from Banič et al. (2022).

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Minimal dynamical systems with closed relations

Abstract: We introduce dynamical systems (X,G) with closed relations G on compact metric spaces X and discuss different types of minimality of such dynamical systems, all of them generalizing minimal dynamical systems (X, f ) with continuous function f on a compact metric space X.

Cited by 3 publications

References 1 publication

Revisiting variations in topological transitivity

Revisiting variations in topological transitivity

Transitive points in CR-dynamical systems

Towards the Complete Classification of the Two Lines Inverse Limit

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