Complexity theory of real functions
Complexity theory of real functions
Computability with low-dimensional dynamical systems
Theoretical Computer Science
Reachability analysis of dynamical systems having piecewise-constant derivatives
Theoretical Computer Science - Special issue on hybrid systems
Complexity and real computation
Complexity and real computation
Theoretical Computer Science
Closed-form analytic maps in one and two dimensions can simulate universal Turing machines
Theoretical Computer Science - Special issue on real numbers and computers
Computability on subsets of Euclidean space I: closed and compact subsets
Theoretical Computer Science - Special issue on computability and complexity in analysis
An effective Riemann mapping theorem
Theoretical Computer Science - Special issue on computability and complexity in analysis
Computable analysis: an introduction
Computable analysis: an introduction
The computational complexity of some julia sets
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
On the Complexity of Real Functions
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Filled Julia Sets with Empty Interior Are Computable
Foundations of Computational Mathematics
Computability of Julia Sets
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)
Dimensions of Points in Self-similar Fractals
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Noise vs computational intractability in dynamics
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
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While most polynomial Julia sets are computable, it has been recently shown [12] that there exist non-computable Julia sets. The proof was non-constructive, and indeed there were doubts as to whether specific examples of parameters with non-computable Julia sets could be constructed. It was also unknown whether the non-computability proof can be extended to the filled Julia sets. In this paper we give an answer to both of these questions, which were the main open problems concerning the computability of polynomial Julia sets. We show how to construct a specific polynomial with a non-computable Julia set. In fact, in the case of Julia sets of quadratic polynomials we give a precise characterization of Juliasets with computable parameters. Moreover, assuming a widely believed conjecture in Complex Dynamics, we give a poly-time algorithm forcomputing a number c such that the Julia set Jz2+c z is non-computable. In contrast with these results, we show that the filled Julia set of a polynomial is always computable.