Invariant tori for the Nos\'e Thermostat near the High-Temperature Limit

Let H(q,p) = p^2/2 + V(q) be a 1-degree of freedom mechanical Hamiltonian with a C^n periodic potential V where n>4. The Nos\'e-thermostated system associated to H is shown to have invariant tori near the infinite temperature limit. This is shown to be true for all thermostats similar to Nos\'e's. These results complement the result of Legoll, Luskin and Moeckel who proved the existence of such tori near the decoupling limit.