1,398 research outputs found
The backward {\lambda}-Lemma and Morse filtrations
Consider the infinite dimensional hyperbolic dynamical system provided by the
(forward) heat semi-flow on the loop space of a closed Riemannian manifold M.
We use the recently discovered backward {\lambda}-Lemma and elements of Conley
theory to construct a Morse filtration of the loop space whose cellular
filtration complex represents the Morse complex associated to the downward
L2-gradient of the classical action functional. This paper is a survey. Details
and proofs will be given in [6].Comment: Conference proceedings, 9 pages, 5 figures. v2: typos corrected,
minor modification
Compactification, topology change and surgery theory
We study the process of compactification as a topology change. It is shown
how the mediating spacetime topology, or cobordism, may be simplified through
surgery. Within the causal Lorentzian approach to quantum gravity, it is shown
that any topology change in dimensions may be achieved via a causally
continuous cobordism. This extends the known result for 4 dimensions.
Therefore, there is no selection rule for compactification at the level of
causal continuity. Theorems from surgery theory and handle theory are seen to
be very relevant for understanding topology change in higher dimensions.
Compactification via parallelisable cobordisms is particularly amenable to
study with these tools.Comment: 1+19 pages. LaTeX. 9 associated eps files. Discussion of disconnected
case adde
Morse homology for the heat flow
We use the heat flow on the loop space of a closed Riemannian manifold to
construct an algebraic chain complex. The chain groups are generated by
perturbed closed geodesics. The boundary operator is defined in the spirit of
Floer theory by counting, modulo time shift, heat flow trajectories that
converge asymptotically to nondegenerate closed geodesics of Morse index
difference one.Comment: 89 pages, 3 figure
Geometry of Universal Magnification Invariants
Recent work in gravitational lensing and catastrophe theory has shown that
the sum of the signed magnifications of images near folds, cusps and also
higher catastrophes is zero. Here, it is discussed how Lefschetz fixed point
theory can be used to interpret this result geometrically. It is shown for the
generic case as well as for elliptic and hyperbolic umbilics in gravitational
lensing.Comment: RevTEX4, 13 pages, submitted to J. Math. Phy
A two-cocycle on the group of symplectic diffeomorphisms
We investigate a two-cocycle on the group of symplectic diffeomorphisms of an
exact symplectic manifolds defined by Ismagilov, Losik, and Michor and
investigate its properties. We provide both vanishing and non-vanishing results
and applications to foliated symplectic bundles and to Hamiltonian actions of
finitely generated groups.Comment: 16 pages, no figure
Quantum Degenerate Systems
Degenerate dynamical systems are characterized by symplectic structures whose
rank is not constant throughout phase space. Their phase spaces are divided
into causally disconnected, nonoverlapping regions such that there are no
classical orbits connecting two different regions. Here the question of whether
this classical disconnectedness survives quantization is addressed. Our
conclusion is that in irreducible degenerate systems --in which the degeneracy
cannot be eliminated by redefining variables in the action--, the
disconnectedness is maintained in the quantum theory: there is no quantum
tunnelling across degeneracy surfaces. This shows that the degeneracy surfaces
are boundaries separating distinct physical systems, not only classically, but
in the quantum realm as well. The relevance of this feature for gravitation and
Chern-Simons theories in higher dimensions cannot be overstated.Comment: 18 pages, no figure
A Svarc-Milnor lemma for monoids acting by isometric embeddings
We continue our programme of extending key techniques from geometric group
theory to semigroup theory, by studying monoids acting by isometric embeddings
on spaces equipped with asymmetric, partially-defined distance functions. The
canonical example of such an action is a cancellative monoid acting by
translation on its Cayley graph. Our main result is an extension of the
Svarc-Milnor Lemma to this setting.Comment: 11 page
Transverse instability for non-normal parameters
We consider the behaviour of attractors near invariant subspaces on varying a
parameter that does not preserve the dynamics in the invariant subspace but is
otherwise generic, in a smooth dynamical system. We refer to such a parameter
as ``non-normal''. If there is chaos in the invariant subspace that is not
structurally stable, this has the effect of ``blurring out'' blowout
bifurcations over a range of parameter values that we show can have positive
measure in parameter space.
Associated with such blowout bifurcations are bifurcations to attractors
displaying a new type of intermittency that is phenomenologically similar to
on-off intermittency, but where the intersection of the attractor by the
invariant subspace is larger than a minimal attractor. The presence of distinct
repelling and attracting invariant sets leads us to refer to this as ``in-out''
intermittency. Such behaviour cannot appear in systems where the transverse
dynamics is a skew product over the system on the invariant subspace.
We characterise in-out intermittency in terms of its structure in phase space
and in terms of invariants of the dynamics obtained from a Markov model of the
attractor. This model predicts a scaling of the length of laminar phases that
is similar to that for on-off intermittency but which has some differences.Comment: 15 figures, submitted to Nonlinearity, the full paper available at
http://www.maths.qmw.ac.uk/~eo
Thurston equivalence of topological polynomials
We answer Hubbard's question on determining the Thurston equivalence class of
``twisted rabbits'', i.e. images of the ``rabbit'' polynomial under n-th powers
of the Dehn twists about its ears.
The answer is expressed in terms of the 4-adic expansion of n. We also answer
the equivalent question for the other two families of degree-2 topological
polynomials with three post-critical points.
In the process, we rephrase the questions in group-theoretical language, in
terms of wreath recursions.Comment: 40 pages, lots of figure
Bott periodicity and stable quantum classes
We use Bott periodicity to relate previously defined quantum classes to
certain "exotic Chern classes" on . This provides an interesting
computational and theoretical framework for some Gromov-Witten invariants
connected with cohomological field theories. This framework has applications to
study of higher dimensional, Hamiltonian rigidity aspects of Hofer geometry of
, one of which we discuss here.Comment: prepublication versio
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