Abstract

Most of known examples of completely integrable systems have zero topological entropy. To investigate their complexity it is therefore necessary to introduce more refined invariants. One first natural attempt is to consider the polynomial growth rate of the covering numbers associated with the iterates of the time-one maps, which we call the freedom index of the system. This index has two major drawbacks: it does not satisfy the variational principle (which states that the toplogical entropy is the supremum of the metric entropies) and does not admit a $\sigma$-union property (which states that the topological entropy on a countable union of invariant domains is the supremum of the entropies on the domains). As a consequence its behaviour is quite dificult to deal with, even for very simple systems.
We introduce another invariant, the weak freedom index, the definition of which is based on Pesin′s dynamical dimension theory and thermodynamic formalism, which is closely related to the freedo index and which moreover posesses the $\sigma$-union property. The stricking fact is that the two indices generally take different values, even for very simple gradient systems on a line, but do coincide for Hamiltonian systems in action-angle form. We take advantage of this property to show how to compute both indices of some classes of integrable systems. We finally show how to extend the scope of the usual dynamical complexity theory to form a hierarchy of new invariants which are relevant in the description of integrable Hamiltonian systems.