We show an invariant Harnack inequality for a class of hypoelliptic ultraparabolic operators with underlying homogeneous Lie group structures. As a byproduct we prove a Liouville type theorem for the related “stationary” operators. We also introduce a notion of link of homogeneous Lie Groups that allows us to show that our results apply to wide classes of operators

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We consider degenerate parabolic equations of the form ∂tu = Δλu + f(u) u|∂Ω = 0, u|t=0 = u0 in a bounded domain Ω ⊂N, where Δλ is a subelliptic operator of the type (Formula presented.) We prove global existence of solutions and characterize their longtime behavior. In particular, we show the existence and finite fractal dimension of the global attractor of the generated semigroup and the convergence of solutions to an equilibrium solution when time tends to infinity