On the computability of rotation sets and their entropies

Burr, Michael A.; Schmoll, Martin; Wolf, Christian

doi:10.1017/etds.2018.45

Cited by 9 publications

(25 citation statements)

References 58 publications

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“…The present work can be seen as part of a recent research trend in which dynamical systems are studied from a computational complexity point of view [6,3,5,15,21,11,2,27,24,7,8]. The general idea is to understand how the computability properties of the main invariants that describe a given system are related to its dynamical, geometrical and analytical properties.…”

Section: Introductionmentioning

confidence: 99%

On the computability properties of topological entropy: a general approach

Gangloff,

Herrera,

Rojas

et al. 2019

Preprint

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The dynamics of symbolic systems, such as multidimensional subshifts of finite type or cellular automata, are known to be closely related to computability theory. In particular, the appropriate tools to describe and classify topological entropy for this kind of systems turned out to be of computational nature. Part of the great importance of these symbolic systems relies on the role they have played in understanding more general systems over non-symbolic spaces. The aim of this article is to investigate topological entropy from a computability point of view in this more general, not necessarily symbolic setting. In analogy to effective subshifts, we consider computable maps over effective compact sets in general metric spaces, and study the computability properties of their topological entropies. We show that even in this general setting, the entropy is always a Σ2-computable number. We then study how various dynamical and analytical constrains affect this upper bound, and prove that it can be lowered in different ways depending on the constraint considered. In particular, we obtain that all Σ2-computable numbers can already be realized within the class of surjective computable maps over {0, 1} N , but that this bound decreases to Π1(or upper)-computable numbers when restricted to expansive maps. On the other hand, if we change the geometry of the ambient space from the symbolic {0, 1} N to the unit interval [0, 1], then we find a quite different situation -we show that the possible entropies of computable systems over [0, 1] are exactly the Σ1(or lower)-computable numbers and that this characterization switches down to precisely the computable numbers when we restrict the class of system to the quadratic family.

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Section: Introductionmentioning

confidence: 99%

On the computability properties of topological entropy: a general approach

Gangloff,

Herrera,

Rojas

et al. 2019

Preprint

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“…[HdMS16], and of thermodynamic invariants (see e.g. in [HM10,BW18]). The computational complexity of individual trajectories in Hamiltonian dynamics has been addressed in e.g.…”

Section: Introductionmentioning

confidence: 99%

How to lose at Monte Carlo: a simple dynamical system whose typical statistical behavior is non computable

Rojas¹,

Yampolsky²

2019

Preprint

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We consider the simplest non-linear discrete dynamical systems, given by the logistic maps fa(x) = ax(1 − x) of the interval [0, 1]. We show that there exist real parameters a ∈ (0, 4) for which almost every orbit of fa has the same statistical distribution in [0, 1], but this limiting distribution is not Turing computable. In particular, the Monte Carlo method cannot be applied to study these dynamical systems.2010 Mathematics Subject Classification. 68Q17 and 37E05.

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“…As a word of caution, particular dynamical objects might not be computable, like entropies associated with some rotations [8]. Examples of topological incomputability also exist, for instance, when determining whether or not the L 2 -cohomology vanishes [20].…”

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confidence: 99%

On computational Poisson geometry I: Symbolic foundations

Evangelista-Alvarado¹,

Ruíz-Pantaleón²,

Suárez–Serrato³

2021

JGM

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<p style='text-indent:20px;'>We present a computational toolkit for (local) Poisson-Nijenhuis calculus on manifolds. Our Python module $\textsf{PoissonGeometry}$ implements our algorithms and accompanies this paper. Examples of how our methods can be used are explained, including gauge transformations of Poisson bivector in dimension 3, parametric Poisson bivector fields in dimension 4, and Hamiltonian vector fields of parametric families of Poisson bivectors in dimension 6.</p>

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On the computability of rotation sets and their entropies

Cited by 9 publications

References 58 publications

On the computability properties of topological entropy: a general approach

On the computability properties of topological entropy: a general approach

How to lose at Monte Carlo: a simple dynamical system whose typical statistical behavior is non computable

On computational Poisson geometry I: Symbolic foundations

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