71 research outputs found
A constructive Borel-Cantelli Lemma. Constructing orbits with required statistical properties
In the general context of computable metric spaces and computable measures we
prove a kind of constructive Borel-Cantelli lemma: given a sequence
(constructive in some way) of sets with effectively summable measures,
there are computable points which are not contained in infinitely many .
As a consequence of this we obtain the existence of computable points which
follow the \emph{typical statistical behavior} of a dynamical system (they
satisfy the Birkhoff theorem) for a large class of systems, having computable
invariant measure and a certain ``logarithmic'' speed of convergence of
Birkhoff averages over Lipshitz observables. This is applied to uniformly
hyperbolic systems, piecewise expanding maps, systems on the interval with an
indifferent fixed point and it directly implies the existence of computable
numbers which are normal with respect to any base.Comment: Revised version. Several results are generalize
Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems
A pseudorandom point in an ergodic dynamical system over a computable metric
space is a point which is computable but its dynamics has the same statistical
behavior as a typical point of the system.
It was proved in [Avigad et al. 2010, Local stability of ergodic averages]
that in a system whose dynamics is computable the ergodic averages of
computable observables converge effectively. We give an alternative, simpler
proof of this result.
This implies that if also the invariant measure is computable then the
pseudorandom points are a set which is dense (hence nonempty) on the support of
the invariant measure
Computability of probability measures and Martin-Lof randomness over metric spaces
In this paper we investigate algorithmic randomness on more general spaces
than the Cantor space, namely computable metric spaces. To do this, we first
develop a unified framework allowing computations with probability measures. We
show that any computable metric space with a computable probability measure is
isomorphic to the Cantor space in a computable and measure-theoretic sense. We
show that any computable metric space admits a universal uniform randomness
test (without further assumption).Comment: 29 page
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
Statistical properties of dynamical systems - simulation and abstract computation.
International audienceWe survey an area of recent development, relating dynamics to theoretical computer science. We discuss some aspects of the theoretical simulation and computation of the long term behavior of dynamical systems. We will focus on the statistical limiting behavior and invariant measures. We present a general method allowing the algorithmic approximation at any given accuracy of invariant measures. The method can be applied in many interesting cases, as we shall explain. On the other hand, we exhibit some examples where the algorithmic approximation of invariant measures is not possible. We also explain how it is possible to compute the speed of convergence of ergodic averages (when the system is known exactly) and how this entails the computation of arbitrarily good approximations of points of the space having typical statistical behaviour (a sort of constructive version of the pointwise ergodic theorem)
Weihrauch-completeness for layerwise computability
We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoff's theorem. We also consider hitting time operators, which share the Weihrauch degree of the former examples but fail to be layerwise computable
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