26 research outputs found
Superrigid subgroups and syndetic hulls in solvable Lie groups
This is an expository paper. It is not difficult to see that every group
homomorphism from the additive group Z of integers to the additive group R of
real numbers extends to a homomorphism from R to R. We discuss other examples
of discrete subgroups D of connected Lie groups G, such that the homomorphisms
defined on D can ("virtually") be extended to homomorphisms defined on all of
G. For the case where G is solvable, we give a simple proof that D has this
property if it is Zariski dense. The key ingredient is a result on the
existence of syndetic hulls.Comment: 17 pages. This is the final version that will appear in the volume
"Rigidity in Dynamics and Geometry," edited by M. Burger and A. Iozzi
(Springer, 2002
Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects
We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation
in phase space. We demonstrate that it accommodates the phase space
dynamics of low dimensional dissipative systems such as the much studied Lorenz
and R\"{o}ssler Strange attractors, as well as the more recent constructions of
Chen and Leipnik-Newton. The rotational, volume preserving part of the flow
preserves in time a family of two intersecting surfaces, the so called {\em
Nambu Hamiltonians}. They foliate the entire phase space and are, in turn,
deformed in time by Dissipation which represents their irrotational part of the
flow. It is given by the gradient of a scalar function and is responsible for
the emergence of the Strange Attractors.
Based on our recent work on Quantum Nambu Mechanics, we provide an explicit
quantization of the Lorenz attractor through the introduction of
Non-commutative phase space coordinates as Hermitian matrices in
. They satisfy the commutation relations induced by one of the two
Nambu Hamiltonians, the second one generating a unique time evolution.
Dissipation is incorporated quantum mechanically in a self-consistent way
having the correct classical limit without the introduction of external degrees
of freedom. Due to its volume phase space contraction it violates the quantum
commutation relations. We demonstrate that the Heisenberg-Nambu evolution
equations for the Quantum Lorenz system give rise to an attracting ellipsoid in
the dimensional phase space.Comment: 35 pages, 4 figures, LaTe