Hans Koch and Sasa Kocic
A renormalization approach to lower-dimensional tori with Brjuno frequency vectors
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ABSTRACT. We extend the renormalization scheme for vector fields on $\mathbb{T}^d\times\mathbb{R}^m$, in order to construct lower-dimensional invariant tori with Brjuno frequency vectors for near-integrable Hamiltonian flows. For every Brjuno frequency vector $\omega\in\mathbb{R}^d$ and every vector $\Omega\in\mathbb{R}^D$ satisfying an arithmetic condition with respect to $\omega$, there exists an analytic manifold $\mathcal{W}$ of infinitely renormalizable Hamiltonian vector fields; each vector field on $\mathcal{W}$ is shown to have an analytic invariant torus with frequency vector $\omega$.