Physics and Computation
2009

!! Talk from Mathieu Hoyrup

!!! Title

Dynamical systems: unpredictability vs incomputability.

!!! Abstract

The long-term forecasting of chaotic dynamical systems is generally
not possible, because of the so-called "sensitivity to initial
conditions": if you do not know exactly the initial state of a system,
you cannot predict its long-range evolution. For this reason, the
mathematical theory of dynamical systems is more focused on the
understanding of the global, asymptotic behaviour of dynamical systems
than the prediction of individual trajectories. In particular, several
notions of entropies of a system are defined in order to quantify its
unpredictability.

Computability theory provides another perspective on the problem of
unpredictability: a system is unpredictable if its evolution cannot be
computed by a machine. Unpredictability is then understood as
incomputability, or difficulty to compute.

I will present some works that connect these two approaches: the
classical one from the mathematical theory of dynamical systems, the
recursion-theoretic one using concepts from computability theory,
especially Kolmogorov complexity.