Towards the complete classification of generalized tent maps inverse limits

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

doi:10.1016/j.topol.2012.09.017

Cited by 15 publications

(10 citation statements)

References 33 publications

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“…They are a generalisation of standard inverse limits and were introduced in [4,5] by Ingram and Mahavier. The concept of these generalised inverse limits has become very popular since their introduction and has been studied by many authors; see [1,3] where more references can be found.…”

Section: Definitions and Notationmentioning

confidence: 99%

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Cardinality of Inverse Limits With Upper Semicontinuous Bonding Functions

Roškarič¹,

Tratnik²

2014

Bull. Aust. Math. Soc.

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We explore the cardinality of generalised inverse limits. Among other things, we show that, for any $n\in \{ℵ_{0},c,1,2,3,\dots \}$, there is an upper semicontinuous function with the inverse limit having exactly $n$ points. We also prove that if $f$ is an upper semicontinuous function whose graph is a continuum, then the cardinality of the corresponding inverse limit is either 1, $ℵ_{0}$ or $c$. This generalises the recent result of I. Banič and J. Kennedy, which claims that the same is true in the case where the graph is an arc.

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Section: Definitions and Notationmentioning

confidence: 99%

“…• the first connects (0, 1 3 ) to ( 1 3 , 1 3 ); • the second connects ( 1 3 , 1 3 ) to (0, 1 2 ); • the third connects (0, 1 2 ) to ( We have seen in Theorem 3.9 and Example 3.10 that | lim ← − − f | ∈ {1, ℵ 0 , c} if G( f ) is connected. If G( f ) is not connected, other possibilities may occur.…”

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confidence: 99%

Cardinality of Inverse Limits With Upper Semicontinuous Bonding Functions

Roškarič¹,

Tratnik²

2014

Bull. Aust. Math. Soc.

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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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“…Generalised inverse limits of compacta were introduced by Ingram and Mahavier in 2006 in [1] and have since received much attention (e.g., [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]). Recall that an inverse limit of a sequence…”

Section: Introductionmentioning

confidence: 99%

Mahavier completeness and classifying diagrams

Maehara

Weiss

2017

Topology and its Applications

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Generalised inverse limits of compacta were introduced by Ingram and Mahavier in 2006. The main difference between ordinary inverse limits and their generalised cousins is that the former concerns diagrams of singlevalued functions while the latter permits multivalued functions. However, generalised inverse limits are not merely limits in the Kleisli category of a hyperspace monad, a fact that independently motivated each of the authors of this article to come up with the same formalism which restores the link with category theory through the concept of Mahavier limit of an order diagram in an order extension of a category B. Mahavier limits of diagrams in B coincide with ordinary limits in B, and so Mahavier limits are an extension of ordinary limits along the functor that views an ordinary diagram as a diagram in the extension. Within that context it is natural to consider Mahavier completeness, namely when all small diagrams admit Mahavier limits, as well as classifying diagrams, namely the existence of a right adjoint to the mentioned functor on diagrams. In this work we show that these two conditions are equivalent, and we study some of the properties of classifying diagrams and of the adjunction.

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Towards the complete classification of generalized tent maps inverse limits

Cited by 15 publications

References 33 publications

Cardinality of Inverse Limits With Upper Semicontinuous Bonding Functions

Cardinality of Inverse Limits With Upper Semicontinuous Bonding Functions

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

Mahavier completeness and classifying diagrams

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