Limits of inverse limits

Banič, Iztok; Črepnjak, Matevž; Merhar, Matej; Milutinović, Uroš

doi:10.1016/j.topol.2009.10.002

Cited by 13 publications

(7 citation statements)

References 2 publications

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“…(p = 1/2) Suppose p = 1/2, but as before J 0 is the vertical line with π 0 (J 0 ) = { p} and J 1 is the line given by x = 1 − y. Note that when p = 0, we get the inverse limit M that has been called the 'Monster' by Banic [BCMM1]. Now the topology and dynamics become more complicated.…”

Section: Dynamical Properties Of Shift Mapsmentioning

confidence: 95%

“…Indeed, the continuum ∞ i=1 G −1 0 = M = lim ← − f 0 has been given the name the 'Monster' by Banič. Several authors have studied it, including Banič [BCMM1] and Ingram [I5]. It contains an invariant Cantor set, is treelike, and has both complicated dynamics and topology.…”

Section: Proposition 1 the Space M Consists Of The Union Of All Arcs Amentioning

confidence: 99%

“…The topic caught the interest of a number of continuum theorists, and is currently an intensely studied area, with papers from many authors at this point. See [B,BCMM1,BCMM2,BCMM3,BK,CR,GK1,GK2,Ill,N1,N2,N3,N4,V], for example. (This list is far from exhaustive, with the number of papers on generalized inverse limits now over 60.)…”

Section: Introductionmentioning

confidence: 99%

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Dynamical properties of shift maps on inverse limits with a set valued function

Kennedy

Nall

2016

Ergod. Th. Dynam. Sys.

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Set-valued functions from an interval into the closed subsets of an interval arise in various areas of science and mathematical modeling. Research has shown that the dynamics of a single-valued function on a compact space are closely linked to the dynamics of the shift map on the inverse limit with the function as the sole bonding map. For example, it has been shown that with Devaney’s definition of chaos the bonding function is chaotic if and only if the shift map is chaotic. One reason for caring about this connection is that the shift map is a homeomorphism on the inverse limit, and therefore the topological structure of the inverse-limit space must reflect in its richness the dynamics of the shift map. In the set-valued case there may not be a natural definition for chaos since there is not a single well-defined orbit for each point. However, the shift map is a continuous single-valued function so it together with the inverse-limit space form a dynamical system which can be chaotic in any of the usual senses. For the set-valued case we demonstrate with theorems and examples rich topological structure in the inverse limit when the shift map is chaotic (on certain invariant sets). We then connect that chaos to a property of the set-valued function that is a natural generalization of an important chaos producing property of continuous functions.

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Section: Dynamical Properties Of Shift Mapsmentioning

confidence: 95%

Section: Proposition 1 the Space M Consists Of The Union Of All Arcs Amentioning

confidence: 99%

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Dynamical properties of shift maps on inverse limits with a set valued function

Kennedy

Nall

2016

Ergod. Th. Dynam. Sys.

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“…Generalized inverse limits, or inverse limits with set-valued functions, a subject studied only since 2003 with its introduction by Bill Mahavier, and much subsequent development by Tom Ingram, provide an entirely new way to study multi-valued functions, a way that does not lose information under iteration. (For the interested reader, we recommend the following references: [B], [BCMM1], [BCMM2], [BCMM3], [BK], [CR] [GK1], [GK2], [Il], [IM2], [I1], [I2], [I3], [I4], [I7], [L], [M], [N1], [N2], [N3], [N4], and [V]. This list is far from exhaustive.…”

Section: Introductionmentioning

confidence: 99%

Closed relations with non-zero entropy that generate no periodic points

Banič¹,

Erceg²,

Kennedy³

2022

Preprint

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The paper is motivated by E. Akin's book about dynamical systems and closed relations [A], and by J. Kennedy's and G. Erceg's recent paper about the entropy of closed relations on closed intervals [EK].In present paper, we introduce the entropy of a closed relation G on any compact metric space X and show its basic properties. We also introduce when such a relation G generates a periodic point or finitely generates a Cantor set. Then we show that periodic points, finitely generated Cantor sets, Mahavier products and the entropy of closed relations are preserved by topological conjugations. Among other things, this generalizes the wellknown results about the topological conjugacy of continuous mappings. Finally, we prove a theorem, giving sufficient conditions for a closed relation G on [0, 1] to have a non-zero entropy. Then we present various examples of closed relations G on [0, 1] such that (1) the entropy of G is non-zero, (2) no periodic point or exactly one periodic point is generated by G, and (3) no Cantor set is finitely generated by G.

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“…The topic of generalized inverse limits is currently an intensely studied area of continuum theory, with papers from many authors at this point. See [B], [BCMM1], [BCMM2], [BCMM3], [BK], [CR] [GK1], [GK2], [Il], [IM2], [I1], [I2], [I3], [I4], [I7], [L], [M], [N1], [N2], [N3], [N4], and [V], for example. (This list is far from exhaustive, with the number of papers on generalized inverse limits now over 60.)…”

Section: Introductionmentioning

confidence: 99%

Topological entropy of closed sets in $[0,1]^2$

Erceg,

Kennedy

2017

Preprint

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We generalize the definition of topological entropy due to Adler, Konheim, and McAndrew [AKM] to set-valued functions from a closed subset A of the interval to closed subsets of the interval. We view these set-valued functions, via their graphs, as closed subsets of [0, 1] 2 . We show that many of the topological entropy properties of continuous functions of a compact topological space to itself hold in our new setting, but not all. We also compute the topological entropy of some examples, relate the entropy to other dynamical and topological properties of the examples, and we give an example of a closed subset G of [0, 1] 2 that has 0 entropy but G ∪ {(p, q)}, where (p, q) ∈ [0, 1] 2 \ G, has infinite entropy.

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Limits of inverse limits

Cited by 13 publications

References 2 publications

Dynamical properties of shift maps on inverse limits with a set valued function

Dynamical properties of shift maps on inverse limits with a set valued function

Closed relations with non-zero entropy that generate no periodic points

Topological entropy of closed sets in $[0,1]^2$

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