Some remarks on modified power entropy

Gröger, Maik; Jäger, Tobias

doi:10.1090/conm/669/13425

Cited by 14 publications

(9 citation statements)

References 38 publications

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“…Positive measure of the boundary of a window of a CPS is not a sufficient criterion for positive topological entropy of the associated Delone dynamical system. We note that there even exist irregular Toeplitz flows (and thus irregular model sets) whose word complexity is only linear [11]. A number of further interesting examples are available in the literature.…”

Section: Consequences For Irregular Toeplitz Flowsmentioning

confidence: 93%

“…We note that there even exist irregular Toeplitz flows (and thus irregular model sets) whose word complexity is only linear [11]. A number of further interesting examples are available in the literature.…”

Section: Corollary 1 Positive Measure Of the Boundary Of A Window Ofmentioning

confidence: 93%

“…(3) Oxtoby's original example of a Toeplitz flow with zero entropy in [21] is not uniquely ergodic, and the same is true of the example in [11]. However, there also exist uniquely ergodic irregular Toeplitz flows both with and without positive entropy [25].…”

Section: Corollary 1 Positive Measure Of the Boundary Of A Window Ofmentioning

confidence: 99%

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Toeplitz flows and model sets

Baake¹,

Jäger²,

Lenz³

2016

Bull. London Math. Soc.

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Abstract. We show that binary Toeplitz flows can be interpreted as Delone dynamical systems induced by model sets and analyse the quantitative relations between the respective system parameters. This has a number of immediate consequences for the theory of model sets. In particular, we use our results in combination with special examples of irregular Toeplitz flows from the literature to demonstrate that irregular proper model sets may be uniquely ergodic and do not need to have positive entropy. This answers questions by Schlottmann and Moody.

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Section: Consequences For Irregular Toeplitz Flowsmentioning

confidence: 93%

Section: Corollary 1 Positive Measure Of the Boundary Of A Window Ofmentioning

confidence: 93%

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Toeplitz flows and model sets

Baake¹,

Jäger²,

Lenz³

2016

Bull. London Math. Soc.

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“…Therefore there has been a search for topological invariants that measure the complexity of dynamical systems in the zero entropy regime. Here we mention two ways how to distinguish between zero entropy systems -topological sequence entropy and polynomial entropy (there are other ways, say one can consider various other kinds of 'slow entropies' in a broad sense, see [KatTho], [HK02], [GJ16], [FGJ16] and references therein, or one can consider the entropy dimension [deC97] or topological complexity of the system [BHM00]; for completeness let us also mention that to distinguish between systems with infinite topological entropy one can use the notion of mean topological dimension [LW00]).…”

Section: Introductionmentioning

confidence: 99%

Rigidity and Flexibility of Polynomial Entropy

Roth,

Snoha

2021

Preprint

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We introduce the notion of a one-way horseshoe and show that the polynomial entropy of an interval map is given by one-way horseshoes of iterates of the map, obtaining in such a way an analogue of Misiurewicz's theorem on topological entropy and standard 'two-way' horseshoes. Moreover, if the map is of Sharkovskii type 1 then its polynomial entropy can also be computed by what we call chains of essential intervals. As a consequence we get a rigidity result that if the polynomial entropy of an interval map is finite, then it is an integer. We also describe the possible values of polynomial entropy of maps of all Sharkovskii types. As an application we compute the polynomial entropy of all maps in the logistic family. On the other hand, we show that in the class of all continua the polynomial entropy of continuous maps is very flexible. For every value α ∈ [0, ∞] there is a homeomorphism on a continuum with polynomial entropy α. We discuss also possible values of the polynomial entropy of continuous maps on dendrites.

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“…The investigation of Toeplitz subshifts has a long history. In fact, Toeplitz subshifts have been rediscovered in various contexts and serve as a prime source of (counter)examples, see for instance [JK69,Wil84] or [GJ16]. Our investigation of Toeplitz subshifts is motivated by their utilisation in two seemingly unrelated fields.…”

Section: Introductionmentioning

confidence: 99%

Combinatorics of one-dimensional simple Toeplitz subshifts

Sell¹

2018

Ergod. Th. Dynam. Sys.

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This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the repetitivity function are proven. Moreover, a complete description of the de Bruijn graphs of the subshifts is given. Finally, the Boshernitzan condition is characterised in terms of combinatorial quantities, based on a recent result of Liu and Qu ([LQ11]). Particular simple characterisations are provided for simple Toeplitz subshifts that correspond to the orbital Schreier graphs of the family of Grigorchuk's groups, a class of subshifts that serves as main example throughout the paper.

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Some remarks on modified power entropy

Cited by 14 publications

References 38 publications

Toeplitz flows and model sets

Toeplitz flows and model sets

Rigidity and Flexibility of Polynomial Entropy

Combinatorics of one-dimensional simple Toeplitz subshifts

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