16 research outputs found
On computational complexity of Siegel Julia sets
It has been previously shown by two of the authors that some polynomial Julia
sets are algorithmically impossible to draw with arbitrary magnification. On
the other hand, for a large class of examples the problem of drawing a picture
has polynomial complexity. In this paper we demonstrate the existence of
computable quadratic Julia sets whose computational complexity is arbitrarily
high.Comment: Updated version, to appear in Commun. Math. Phy
Dynamics and abstract computability: computing invariant measures
We consider the question of computing invariant measures from an abstract
point of view. We work in a general framework (computable metric spaces,
computable measures and functions) where this problem can be posed precisely.
We consider invariant measures as fixed points of the transfer operator and
give general conditions under which the transfer operator is (sufficiently)
computable. In this case, a general result ensures the computability of
isolated fixed points and hence invariant measures (in given classes of
"regular" measures). This implies the computability of many SRB measures.
On the other hand, not all computable dynamical systems have a computable
invariant measure. We exhibit two interesting examples of computable dynamics,
one having an SRB measure which is not computable and another having no
computable invariant measure at all, showing some subtlety in this kind of
problems
Computability of Julia sets
In this paper we settle most of the open questions on algorithmic
computability of Julia sets. In particular, we present an algorithm for
constructing quadratics whose Julia sets are uncomputable. We also show that a
filled Julia set of a polynomial is always computable.Comment: Revised. To appear in Moscow Math. Journa