arXiv subjects

Mathieu Hoyrup

Publications and source records attributed to Mathieu Hoyrup.

25 records · Page 2Linked to original sources

arXiv2011

Dynamical systems, simulation, abstract computation

We survey an area of recent development, relating dynamics to theoretical computer science. We discuss the theoretical limits of simulation and computation of interesting quantities in dynamical systems. We will focus on central objects of the theory of dynamics, as invariant measures and invariant sets, showing that even if they can be computed with arbitrary precision in many interesting cases, there exists some cases in which they can not. 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).

math.DS↗

arXiv2010

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.

math.NA↗

arXiv2009

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.

math.DS↗

arXiv2009

Randomness on Computable Probability Spaces - A Dynamical Point of View

We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the statistical behavior of the system (Birkhoff's pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every mixing computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications.

math.PR↗

arXiv2008

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).

cs.IT↗

arXiv2008

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 $A_{i}$ with effectively summable measures, there are computable points which are not contained in infinitely many $A_{i}$. 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.

math.CA↗

arXiv2008

Effective symbolic dynamics, random points, statistical behavior, complexity and entropy

We consider the dynamical behavior of Martin-Löf random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure. We use computable partitions to define a sort of effective symbolic model for the dynamics. Through this construction we prove that such points have typical statistical behavior (the behavior which is typical in the Birkhoff ergodic theorem) and are recurrent. We introduce and compare some notions of complexity for orbits in dynamical systems and prove: (i) that the complexity of the orbits of random points equals the Kolmogorov-Sinaï entropy of the system, (ii) that the supremum of the complexity of orbits equals the topological entropy.

math.DS↗