Topological entropy on closed sets in [0,1] 2

Erceg, Goran; Kennedy, Judy

doi:10.1016/j.topol.2018.06.015

Cited by 12 publications

(19 citation statements)

References 19 publications

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“…We make use of equivalent definitions for topological entropy. Two of them are presented in [8], and the other is presented in [5]. The equivalence of each of these definitions is shown in or is a consequence of [12,Theorem 2.2], [8,Theorem 3.1], and [5,Theorem 4.3].…”

Section: Preliminariesmentioning

confidence: 99%

“…This is especially true when studying inverse limits of continuous functions on the unit interval, such as unimodal maps. Although most work in the area of set-valued inverse limits has focused on the topological aspects (particularly, investigating the various continua that can arise as inverse limits), some recent work has focused on the dynamical aspects and, in particular, the topological entropy, [5,8,9].…”

Section: Introductionmentioning

confidence: 99%

“…The concept of topological entropy was first introduced by Adler, Konheim, and McAndrew [1], and is a measure of the complexity of the dynamics of a function. Recent results in [8] and [5] have generalized the definition of topological entropy to the family of upper semi-continuous set-valued functions. In this paper, we use these generalizations to investigate the topological entropy for a family of set-valued functions, called Markov set-valued functions, that are generalizations of Markov interval maps.…”

Section: Introductionmentioning

confidence: 99%

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Topological entropy of Markov set-valued functions

Alvin¹,

Kelly²

2019

Ergod. Th. Dynam. Sys.

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We investigate the entropy for a class of upper semi-continuous set-valued functions, called Markov set-valued functions, that are a generalization of single-valued Markov interval functions. It is known that the entropy of a Markov interval function can be found by calculating the entropy of an associated shift of finite type. In this paper, we construct a similar shift of finite type for Markov set-valued functions and use this shift space to find upper and lower bounds on the entropy of the set-valued function.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Topological entropy of Markov set-valued functions

Alvin¹,

Kelly²

2019

Ergod. Th. Dynam. Sys.

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“…For example, the function f in [10,Example 2.11] is a Markov-like function and the generalized inverse limit M is the Hurewicz continuum. This example is studied in [8,13,14]. In section 3, we will prove that any two generalized inverse limits with Markov-like bonding functions having the same pattern are homeomorphic.…”

Section: Introductionmentioning

confidence: 99%

Markov-like set-valued functions on intervals and their inverse limits

Imamura¹

2018

Glas. Mat. Ser. III

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We introduce Markov-like functions on intervals as a generalization of generalized Markov interval functions and define the notation of the same pattern between Markov-like functions. Then we show that two generalized inverse limits with Markov-like bonding functions having the same pattern are homeomorphic. This result gives a generalization of the results of S. Holte ([9]) and I. Banič and T. Lunder ([5]). ←− {X n , f n } of the inverse sequence {X n , f n } n∈N is defined by lim ←− {X n , f n } := {(x 1 , x 2 ,. . .) ∈ ∞ n=1 X n | x n ∈ f n (x n+1) for any n ∈ N}.

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“…In the past decade multi-maps have been studied extensively, with a particular focus on the topological structure of the associated space of trajectories or a related inverse limit space; see [15]. This development has also led to a renewed interest in the dynamics of multi-maps [17,18,13,11]. Additionally, multi-maps are the topological analogues of random maps of the interval, which have received substantial attention, e.g., [10,14,23,4].…”

mentioning

confidence: 99%

Entropy conjugacy for Markov multi-maps of the interval

Kelly¹,

McGoff²

2021

Discrete &Amp; Continuous Dynamical Systems - A

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We consider a class F of Markov multi-maps on the unit interval. Any multi-map gives rise to a space of trajectories, which is a closed, shiftinvariant subset of [0, 1] Z +. For a multi-map in F , we show that the space of trajectories is (Borel) entropy conjugate to an associated shift of finite type. Additionally, we characterize the set of numbers that can be obtained as the topological entropy of a multi-map in F .

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Topological entropy on closed sets in [0,1] 2

Cited by 12 publications

References 19 publications

Topological entropy of Markov set-valued functions

Topological entropy of Markov set-valued functions

Markov-like set-valued functions on intervals and their inverse limits

Entropy conjugacy for Markov multi-maps of the interval

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