On Hamiltonian flows whose orbits are straight lines
Hans Koch Héctor E. Lomelí
We consider real analytic Hamiltonians on $\mathbb{R}^n \times \mathbb{R}^n$ whose flow depends linearly on time. Trivial examples are Hamiltonians $H(q,p)$ that do not depend on the coordinate $q\in \mathbb{R}^n$. By a theorem of Moser [11], every polynomial Hamiltonian of degree $3$ reduces to such a $q$-independent Hamiltonian via a linear symplectic change of variables. We show that such a reduction is impossible, in general, for polynomials of degree $4$ or higher. But we give a condition that implies linear-symplectic conjugacy to another simple class of Hamiltonians. The condition is shown to hold for all nondegenerate Hamiltonians that are homogeneous of degree $4$.
keywords: Hamiltonian Iwasawa decomposition symplectic map polynomial map jolt factorization. symplectic matrix
On hyperbolicity in the renormalization of near-critical area-preserving maps
Hans Koch
We consider MacKay's renormalization operator for pairs of area-preserving maps, near the fixed point obtained in [1]. Of particular interest is the restriction $\mathfrak{R}_{0}$ of this operator to pairs that commute and have a zero Calabi invariant. We prove that a suitable extension of $\mathfrak{R}_{0}^{3}$ is hyperbolic at the fixed point, with a single expanding direction. The pairs in this direction are presumably commuting, but we currently have no proof for this. Our analysis yields rigorous bounds on various ``universal'' quantities, including the expanding eigenvalue.
keywords: invariant circle hyperbolicity. renormalization Area-preserving maps
On the renormalization of Hamiltonian flows, and critical invariant tori
Hans Koch
We analyze a renormalization group transformation $\mathcal R$ for partially analytic Hamiltonians, with emphasis on what seems to be needed for the construction of non-integrable fixed points. Under certain assumptions, which are supported by numerical data in the golden mean case, we prove that such a fixed point has a critical invariant torus. The proof is constructive and can be used for numerical computations. We also relate $\mathcal R$ to a renormalization group transformation for commuting maps.
keywords: fixed point. Hamiltonian flows
Renormalization of diophantine skew flows, with applications to the reducibility problem
Hans Koch João Lopes Dias
We introduce a renormalization group framework for the study of quasiperiodic skew flows on Lie groups of real or complex $n\times n$ matrices, for arbitrary Diophantine frequency vectors in $R^{d}$ and dimensions $d,n$. In cases where the Lie algebra component of the vector field is small, it is shown that there exists an analytic manifold of reducible skew systems, for each Diophantine frequency vector. More general near-linear flows are mapped to this case by increasing the dimension of the torus. This strategy is applied for the group of unimodular $2\times 2$ matrices, where the stable manifold is identified with the set of skew systems having a fixed fibered rotation number. Our results apply to vector fields of class Cγ, with $\gamma$ depending on the number of independent frequencies, and on the Diophantine exponent.
keywords: reducibility of skew flows. Key words and phrases: Renormalization of flows
A renormalization group fixed point associated with the breakup of golden invariant tori
Hans Koch
We give a computer-assisted proof for the existence of a renormalization group fixed point (Hamiltonian) with non-trivial scaling, associated with the breakup of invariant tori with rotation number equal to the golden mean.
keywords: breakup of invariant tori computer-assisted proof. Hamiltonian flows
Aubry-Mather theory for functions on lattices
Hans Koch Rafael De La Llave Charles Radin
We generalize the Aubry-Mather theorem on the existence of quasi-periodic solutions of one dimensional difference equations to situations in which the independent variable ranges over more complicated lattices. This is a natural generalization of Frenkel-Kontorova models to physical situations in a higher dimensional space. We also consider generalizations in which the interactions among the particles are not just nearest neighbor, and indeed do not have finite range.
keywords: functions on lattices. Aubry-Mather theory

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