Thermalization in a periodically driven fully connected quantum Ising ferromagnet

means of a Floquet analysis, we study the quantum dynamics of a fully connected Lipkin-Ising ferromagnet in a periodically driven transverse field showing that thermalization in the steady state is intimately connected to properties of the N -> ? classical Hamiltonian dynamics. When the dynamics is ergodic, the Floquet spectrum obeys a Wigner-Dyson statistics and the system satisfies the eigenstate thermalization hypothesis (ETH): Independently of the initial state, local observables relax to the T = ? thermal value, and Floquet states are delocalized in the Hilbert space. On the contrary, if the classical dynamics is regular no thermalization occurs. We further discuss the relationship between ergodicity and dynamical phase transitions, and the relevance of our results to other fully connected periodically driven models (like the Bose-Hubbard one), and possibilities of experimental realization in the case of two coupled BEC.