Fractals everywhere
Theoretical Computer Science
Computable analysis: an introduction
Computable analysis: an introduction
Jordan Curves with Polynomial Inverse Moduli of Continuity
Electronic Notes in Theoretical Computer Science (ENTCS)
Constructing non-computable Julia sets
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Jordan curves with polynomial inverse moduli of continuity
Theoretical Computer Science
Electronic Notes in Theoretical Computer Science (ENTCS)
Complexity of Operators on Compact Sets
Electronic Notes in Theoretical Computer Science (ENTCS)
On the Complexity of Convex Hulls of Subsets of the Two-Dimensional Plane
Electronic Notes in Theoretical Computer Science (ENTCS)
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Dimensions of Points in Self-similar Fractals
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Hyperbolic Julia Sets are Poly-Time Computable
Electronic Notes in Theoretical Computer Science (ENTCS)
A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions
Electronic Notes in Theoretical Computer Science (ENTCS)
Computability of countable subshifts
CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe
On the complexity of computing the logarithm and square root functions on a complex domain
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
How can nature help us compute?
SOFSEM'06 Proceedings of the 32nd conference on Current Trends in Theory and Practice of Computer Science
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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 reliably 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 complex functions f(z)=z2+c for |c|. We will give a strict computable error estimation w.r.t. the Hausdorff metric dH which means that the set is recursive [10]. Although these and many more Julia sets J are proven to be recursive [12] and furthermore recursive compact subsets of the Euclidean plane are known to have a computable Turing machine time complexity [10], hardly anything is known about the computational complexity of non-trivial examples. The algorithms given in this paper compute the Julia sets locally in time O(k2• M(k)) (where M(k) is a time bound for multiplication of two k-bit integers). Roughly speaking, the local time complexity is the number of Turing machine steps to decide for a single point whether it belongs to a grid Kk⊆ (2-k• Z)2 such that dH(Kk,J)≤ 2-k.