Almost Every Real Quadratic Polynomial has a Poly-time Computable Julia Set

Dudko, Artem; Yampolsky, Michael

doi:10.1007/s10208-017-9367-7

Cited by 3 publications

(4 citation statements)

References 19 publications

(13 reference statements)

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“…It not only has zero area, but, subject to a natural conjecture, also has zero Hausdorff dimension. It is plausible that for a typical value of c, the set J c is computable in polynomial time, and this is indeed true for a typical real value of c as shown by A. Dudko and I in [10].…”

Section: Can One Compute An Attractor As a Set?mentioning

confidence: 89%

Towards understanding the theoretical challenges of numerical modeling of dynamical systems

Yampolsky

2021

NZ J Math

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Section: Can One Compute An Attractor As a Set?mentioning

confidence: 89%

Towards understanding the theoretical challenges of numerical modeling of dynamical systems

Yampolsky

2021

NZ J Math

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“…This means that poly-time computability is a "physically natural" property in real dynamics. Conjecturally, the main technical result of [26] should imply the same statement for complex parameters c as well, but the conjecture in question (Collet-Eckmann parameters form a set of full measure among non-hyperbolic parameters) while long-established, is stronger than Density of Hyperbolicity Conjecture, and is currently out of reach.…”

Section: Both Figures Courtesy Of Arnaud Chéritatmentioning

confidence: 94%

“…Let us further specialize to real quadratic family f c , c ∈ R. In this case, it was recently proved by Dudko and Yampolsky [26] that: Theorem 5.14. Almost every real quadratic Julia set is poly-time.…”

Section: Both Figures Courtesy Of Arnaud Chéritatmentioning

confidence: 99%

Computable geometric complex analysis and complex dynamics

Rojas¹,

Yampolsky²

2017

Preprint

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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 [5,3,6,16,23,12,2,29,26,21,7,8]. 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.…”

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confidence: 99%

Computability of topological entropy: From general systems to transformations on Cantor sets and the interval

Gangloff

Herrera

Rojas

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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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 algorithmic. 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 Σ 2 -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 Σ 2 -computable numbers can already be realized within the class of surjective computable maps over {0, 1} N , but that this bound decreases to Π 1 (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 Σ 1 (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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Almost Every Real Quadratic Polynomial has a Poly-time Computable Julia Set

Abstract: We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time.arXiv:1702.05768v2 [math.DS]

Cited by 3 publications

References 19 publications

Towards understanding the theoretical challenges of numerical modeling of dynamical systems

Towards understanding the theoretical challenges of numerical modeling of dynamical systems

Computable geometric complex analysis and complex dynamics

Computability of topological entropy: From general systems to transformations on Cantor sets and the interval

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