The computational complexity of some julia sets

Rettinger, Robert; Weihrauch, Klaus

doi:10.1145/780569.780570

Cited by 9 publications

(10 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Roughly speaking, the local time complexity is the number of Turing machine steps to give a (zoomed) set of pixels with Hausdorff distance at most 2 −k+2 . This gives the first polynomial bound on the complexity of such a general and rich class of Julia sets and extends a result of Rettinger and Weihrauch in [10], where a similar algorithm is given for polynomials z → z 2 + c for |c| < 1/4. Furthermore, in [12] it is shown that the Julia sets of hyperbolic polynomials are recursive.…”

Section: Introductionsupporting

confidence: 71%

“…The local complexity of subsets of C was introduced in [10]. Given a Turing machine M, let L(M) denote the set accepted by M and let U M (k) be the union of all intervals I k,i,j = [2 −k · i; 2 −k · (i + 1)] × [2 −k · j; 2 −k · (j + 1)] with "0 k , i, j" ∈ L(M), where i, j are given by their binary representation.…”

Section: Local Complexity On C ∞mentioning

confidence: 99%

“…Furthermore we fix ε := 1. For compact subsets of C the local complexity (as subset of C ∞ ) and the complexity defined in [10] coincide. Thus again we could draw parts of the approximations according to the discussion given above, i.e.…”

Section: Local Complexity On C ∞mentioning

confidence: 99%

“…In Section 3 we mainly discuss extensions to the local repelling property of J. This result allows us to use similar ideas to [10] even in the general case of rational functions. In Section 4 we then give and discuss the definition of local complexity on C ∞ .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions

Rettinger

2005

Electronic Notes in Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

Although numerous computer programs have been written to compute sets of points which claim to approximate Julia sets, no reliable high precision pictures of non-trivial Julia sets are currently known. Usually, no error estimates are added and even those algorithms which work reliable in theory, become unreliable in practice due to rounding errors and the use of fixed length floating point numbers. In this paper we prove the existence of polynomial time algorithms to approximate the Julia sets of given hyperbolic rational functions. We will give a strict computable error estimation w.r.t. the Hausdorff metric on the complex sphere. This extends a result on polynomials z → z 2 + c, where |c| < 1/4, in [10] and an earlier result in [12] on the recursiveness of the Julia sets of hyperbolic polynomials. The algorithm given in this paper computes Julia sets locally in time O(k · M (k)) (where M (k) denotes the time needed to multiply two k-bit numbers). Roughly speaking, the local time complexity is the number of Turing machine steps to decide a set of disks of spherical diameter 2 −k so that the union of these disks has Hausdorff distance at most 2 −k+2 . This allows to give reliable pictures of Julia sets to arbitrary precision.

show abstract

Section: Introductionsupporting

confidence: 71%

Section: Local Complexity On C ∞mentioning

confidence: 99%

Section: Local Complexity On C ∞mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions

Rettinger

2005

Electronic Notes in Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…The computability of complex dynamical systems has been investigated by Rettinger and Weihrauch [18] and by Braverman and Yampolsky [2]. The study of the computability of dynamical systems has received increasing attention in recent years; see for example papers of Delvenne et al [12], Hochman [13], Jeandal and Vanier [24], Miller [17] and Simpson [21].…”

mentioning

confidence: 99%

Computability of Countable Subshifts in One Dimension

et al. 2011

View full text Add to dashboard Cite

We investigate the computability of countable subshifts in one dimension, and their members. Subshifts of Cantor-Bendixson rank two contain only eventually periodic elements. Any rank two subshift in 2 Z is decidable. Subshifts of rank three may contain members of arbitrary Turing degree. In contrast, effectively closed ( 0 1 ) subshifts of rank three contain only computable elements, but 0 1 subshifts of rank four may contain members of arbitrary 0 2 degree. There is no subshift of rank ω + 1.

show abstract

Computability and Dynamical Systems

Buescu

Graça

Zhong

2011

Dynamics, Games and Science I

View full text Add to dashboard Cite

In this paper we explore results that establish a link between dynamical systems and computability theory (not numerical analysis). In the last few decades, computers have increasingly been used as simulation tools for gaining insight into dynamical behavior. However, due to the presence of errors inherent in such numerical simulations, with few exceptions, computers have not been used for the nobler task of proving mathematical results. Nevertheless, there have been some recent developments in the latter direction. Here we introduce some of the ideas and techniques used so far, and suggest some lines of research for further work on this fascinating topic.

show abstract

The computational complexity of some julia sets

Cited by 9 publications

References 0 publications

A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions

A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions

Computability of Countable Subshifts in One Dimension

Computability and Dynamical Systems

Contact Info

Product

Resources

About