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DCDS

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

DCDS

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.

DCDS

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.

DCDS

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 C

^{γ}, with $\gamma$ depending on the number of independent frequencies, and on the Diophantine exponent.
DCDS

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.

DCDS

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.

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