ABSTRACT:

Examples of near-integrable Hamiltonian systems with $N+\frac{1}{2}$ degrees of freedom ($N\ge2$), which are as smooth as possible and as ``unstable'' as possible are constructed.
They are generated by Hamiltonian functions of the form
$\frac{1}{2}(I_1^2+\cdots+I_N^2) + f(\theta,I,t)$
with Gevrey functions~$f$ which are $1$-periodic in time and arbitrarily small.
The Nekhoroshev theorem forces exponential stability.
Still, for these particular~$f$, one can find probability measures in the phase space such that the first $N-1$ action variables satisfy a functional central limit theorem: when properly rescaled, they converge in law to a Brownian motion (with an exponentially small coefficient of diffusion, as should be). These probability measures are related to the product of the $2(N-1)$-fold Lebesgue measure by a measure supported on a horseshoe on the last degree of freedom.
There is a related invariant measure~$\mu$ for the time-$1$ map, which is $\sigma$-finite but not finite. One can show that, when $N=1$ or $N=2$, the time-$1$ map is $\mu$-ergodic. For $N\ge3$, $\mu$-almost orbit is biasymptotic to infinity and the system is not ergodic (but there still exist orbits which are dense in the support of~$\mu$, which is the product of the first $N-1$ degrees of freedom by the horseshoe).