Section: New Results

Dynamical systems

Participant : Mathieu Hoyrup.

Birkhoff theorem is a central result in ergodic theory. Consider a dynamical system $(X,T:X\to X)$, start with an initial condition $x\in X$ and construct the trajectory $(x,T\left(x\right),T^2\left(x\right),...)$. How is this trajectory distributed in $X$? What is the limit frequency of visits of a set $A\subseteq X$? Ergodic theorems answer to these questions by showing (i) that the distribution of almost every point converges and (ii) by describing the possible distributions associated to trajectories.

For several years we have been working on the project of identifying the exact computational content of several ergodic theorems: can the speed of convergence of limit frequencies be computed? Can one distinguish between points with different limit frequencies? Can we construct (compute) points whose trajectory follow a prescribed distribution? How random (i.e. incompressible) a point has to be for the distribution of its trajectory to converge?

Limit frequencies

We have obtained new insight in the above questions by proving that random elements eventually reach effective closed sets of positive measure (while it was only known for a more restricted class of sets). The paper appeared in Information and Computation [11] . This result is a key tool in the proof of the result published in [23] .

Information

A chaotic system is unpredictable because it has much more trajectories than observable initial conditions: hence many undistinguishable initial points lead to radically different trajectories. As there are many trajectories, most of them are complex in the sense that they can hardly be compressed, i.e. described in a shorter way than simply listing them. The Shannon-McMilan-Breiman theorem states that the compression-rate of most trajectories coincides with the entropy of the system.

We have been interested in the computational content of this theorem: how random a point has to be to generate a trajectory whose compression rate is the entropy? This question was raised in [71] and has been left open for 14 years. We have solved the problem by showing that Martin-Löf notion of randomness is sufficient. Our recent result presented in [11] is a key ingredient of our proof. We presented the result at STACS [23] .

Decomposition

The ergodic decomposition theorem states that a dynamical system can always be uniquely decomposed into indecomposable subsystems, technically ergodic subsystems. We have been interested in the computability of the decomposition operation. It is known from [71] that this operation is not computable in general. Whether this operation is still not computable when the system can be decomposed into a finite number of subsystems was open. We raised the question and answer it negatively in [57] . More precisely, we prove the existence of ergodic measures $P$ and $Q$ such that neither $P$ nor $Q$ is computable relative to $P+Q$. In other words, the operation of splitting a non-ergodic process into ergodic components is not computable, even in the trivial case of a combination of 2 ergodic processes. The paper is currently in press and will appear in Annals of Pure and Applied Logic [14] .