Filled Julia Sets with Empty Interior Are Computable

Binder, Ilia; Braverman, Mark; Yampolsky, Michael

doi:10.1007/s10208-005-0210-1

Cited by 21 publications

(36 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One can show combining the results of [BBY1] with, for example, those of Petersen and Zakeri [PZ] that this set has Lebesgue measure zero; and Theorem 5.1 implies that its complement contains a dense G δ subset of R/Z. It is natural to ask if, for example, its Hausdorff dimension is positive, and the answer to this question is not known to us.…”

Section: Discussionmentioning

confidence: 99%

Non-computable Julia sets

Braverman

Yampolsky

2005

J. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Discussionmentioning

confidence: 99%

Non-computable Julia sets

Braverman

Yampolsky

2005

J. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…As we showed in [BBY07], a quadratic polynomial with an uncomputable Julia set necessarily possesses a cycle of Siegel disks. Further, we demonstrated in [BBY07]: Theorem 1.…”

Section: Preliminariesmentioning

confidence: 69%

“…The union of all numbers of a bounded type are Diophantine numbers of exponent 2; a zero measure subset of T. As we have shown in [BBY07]:…”

Section: Examples Of Computable and Locally Connected Siegel Julia Sementioning

confidence: 92%

Constructing Locally Connected Non-Computable Julia Sets

Braverman

Yampolsky

2009

Commun. Math. Phys.

View full text Add to dashboard Cite

Abstract. A locally connected quadratic Siegel Julia set has a simple explicit topological model. Such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. We constructively produce parameter values for Siegel quadratics for which the Julia sets are non-computable, yet locally connected. PreliminariesIn this paper, we will assume that the reader is familiar with the concept of computability of a subset of R n and its applications to Julia sets of rational functions. We refer the reader to our paper [BY08a] and the book [BY08b] for an introduction to computability of functions and sets in R n , as it applies to the study of Julia sets. A detailed treatment of computability over the reals is found in [Wei00].We will denote f c (z) = z 2 + c, andz two parameterizations of the quadratic family. The latter is more convenient in studying quadratics with a neutral fixed point. We denote J c , J θ and K c , K θ the Julia sets and the filled Julia sets respectively. Suppose, a polynomial f c has a periodic Siegel disk ∆ centered at a point ζ. Consider a conformal isomorphism φ : D → ∆ mapping 0 to ζ. The conformal radius of the Siegel disk ∆ is the quantity r(∆) = |φ (0)|.A polynomial P θ with θ ∈ R has a neutral fixed point at the origin. When this point is of Siegel type, we denote ∆ θ the Siegel disk around it, and setFor all other values of θ ∈ R we set r(θ) = 0.Informally, the Julia set J c (or J θ ) is computable if, given arbitrarily good approximations of the parameter c (or θ), a Turing Machine can output images of J c (or J θ ) with an arbitrarily high resolution. The parameter is provided to the machine via an oracle, which the machine can query with an arbitrarily high precision. In [BY06] we showed that, surprisingly, there exist parameters c for which the Julia set J c is not computable. In [BY08a] we demonstrated that such parameters can themselves be computed with an arbitrary precision by an explicit algorithm. The practical implications of these results are quite striking: there are computable values of c for which J c cannot be visualized numerically.

show abstract

“…To address the question of computability of J θ for θ ∈ B we first make note of the following result, proven in [BBY1]:…”

Section: Computability Of Julia Sets Of Siegel Quadratics and Negativmentioning

confidence: 99%