SEMINARI DI SISTEMI DINAMICI OLOMORFI (e dintorni)

Mercoledi' 15 novembre ore 14.00

Centro di Ricerca Matematica De Giorgi - Sala Conferenze

David Sauzin (IMCCE)

"Ergodicity and convergence to a Brownian motion in examples of Arnold ì diffusion."

Abstract:

In the same line of research as [Marco-Sauzin, ETDS 2004], I construct examples of near-integrable Hamiltonian systems with $N+1/2$ degrees of freedom ($N\ge2$), which are as smooth as possible and as ``unstable'' as possible. They are generated by Hamiltonian functions of the form $\frac{1}{2}(I_12+\cdots+I_N2) + f(\theta,I,t)$ with Gevrey functions $f$ which are $1$-periodic in time and arbitrarily small. The Nekhoroshev theorem [Marco-Sauzin, IHES 2002] 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 $m$ 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 $m$-ergodic. For $N\ge3$, $m$-almost orbit is biasymptotic to infinity and the system is not ergodic (but there still exist orbits which are dense in the support of $m$, which is the product of the first $N-1$ degrees of freedom by the horseshoe).

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