103 research outputs found
Reeb graph and quasi-states on the two-dimensional torus
This note deals with quasi-states on the two-dimensional torus. Quasi-states
are certain quasi-linear functionals (introduced by Aarnes) on the space of
continuous functions. Grubb constructed a quasi-state on the torus, which is
invariant under the group of area-preserving diffemorphisms, and which moreover
vanishes on functions having support in an open disk. Knudsen asserted the
uniqueness of such a quasi-state; for the sake of completeness, we provide a
proof. We calculate the value of Grubb's quasi-state on Morse functions with
distinct critical values via their Reeb graphs. The resulting formula coincides
with the one obtained by Py in his work on quasi-morphisms on the group of
area-preserving diffeomorphisms of the torus.Comment: 8 page
Orthogonal surfaces
Orthogonal surfaces are nice mathematical objects which have interesting
connections to various fields, e.g., integer programming, monomial ideals and
order dimension. While orthogonal surfaces in one or two dimensions are rather
trivial already the three dimensional case has a rich structure with
connections to Schnyder woods, planar graphs and 3-polytopes.
Our objective is to detect more of the structure of orthogonal surfaces in
four and higher dimensions. In particular we are driven by the question which
non-generic orthogonal surfaces have a polytopal structure.
We study characteristic points and the cp-orders of orthogonal surfaces,
i.e., the dominance orders on the characteristic points. In the generic case
these orders are (almost) face lattices of polytopes. Examples show that in
general cp-orders can lack key properties of face lattices. We investigate
extra requirements which may help to have cp-orders which are face lattices.
Finally, we turn the focus and ask for the realizability of polytopes on
orthogonal surfaces. There are criteria which prevent large classes of
simplicial polytopes from being realizable. On the other hand we identify some
families of polytopes which can be realized on orthogonal surfaces
Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle
Let be a Morse function on a connected compact surface
, and and be respectively the stabilizer
and the orbit of with respect to the right action of the group of
diffeomorphisms . In a series of papers the first author
described the homotopy types of connected components of and
for the cases when is either a -disk or a cylinder or
. Moreover, in two recent papers the authors considered special
classes of smooth functions on -torus and shown that the computations
of for those functions reduces to the cases of -disk
and cylinder.
In the present paper we consider another class of Morse functions
whose KR-graphs have exactly one cycle and prove that for
every such function there exists a subsurface , diffeomorphic
with a cylinder, such that is expressed via the
fundamental group of the restriction of to .
This result holds for a larger class of smooth functions having the following property: for every critical point of
the germ of at is smoothly equivalent to a homogeneous polynomial
without multiple factors.Comment: 17pages, 8 figure
Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles
The main theme of this paper is a relative version of the almost existence
theorem for periodic orbits of autonomous Hamiltonian systems.
We show that almost all low levels of a function on a geometrically bounded
symplectically aspherical manifold carry contractible periodic orbits of the
Hamiltonian flow, provided that the function attains its minimum along a closed
symplectic submanifold. As an immediate consequence, we obtain the existence of
contractible periodic orbits on almost all low energy levels for twisted
geodesic flows with symplectic magnetic field. We give examples of functions
with a sequence of regular levels without periodic orbits, converging to an
isolated, but very degenerate, minimum.
The proof of the relative almost existence theorem hinges on the notion of
the relative Hofer-Zehnder capacity and on showing that this capacity of a
small neighborhood of a symplectic submanifold is finite. The latter is carried
out by proving that the flow of a Hamiltonian with sufficiently large variation
has a non-trivial contractible one-periodic orbit, when the Hamiltonian is
constant and equal to its maximum near a symplectic submanifold and supported
in a neighborhood of the submanifold.Comment: AMS-LaTeX2e, 35 pages, 2 figures. References and new results on
time-dependent flows adde
Slow volume growth for Reeb flows on spherizations and contact Bott--Samelson theorems
We give a uniform lower bound for the polynomial complexity of all Reeb flows
on the spherization (S*M,\xi) over a closed manifold. Our measure for the
dynamical complexity of Reeb flows is slow volume growth, a polynomial version
of topological entropy, and our uniform bound is in terms of the polynomial
growth of the homology of the based loops space of M. As an application, we
extend the Bott--Samelson theorem from geodesic flows to Reeb flows: If
(S*M,\xi) admits a periodic Reeb flow, or, more generally, if there exists a
positive Legendrian loop of a fibre S*_q M, then M is a circle or the
fundamental group of M is finite and the integral cohomology ring of the
universal cover of M is the one of a compact rank one symmetric space.Comment: 34 pages, 1 figur
Dirac operators on cobordisms: degenerations and surgery
We investigate the Dolbeault operator on a pair of pants, i.e., an elementary
cobordism between a circle and the disjoint union of two circles. This operator
induces a canonical selfadjoint Dirac operator on each regular level set
of a fixed Morse function defining this cobordism. We show that as we
approach the critical level set from above and from below these operators
converge in the gap topology to (different) selfadjoint operators that
we describe explicitly. We also relate the Atiyah-Patodi-Singer index of the
Dolbeault operator on the cobordism to the spectral flows of the operators
on the complement of and the Kashiwara-Wall index of a triplet of
finite dimensional lagrangian spaces canonically determined by .Comment: 31 pages, 3 figures; completely rewrote Section 4 using the
definition of Kirk and Lesch of the Kashiwara-Wall inde
Differential algebra of cubic planar graphs
In this article we associate a combinatorial differential graded algebra to a
cubic planar graph G. This algebra is defined combinatorially by counting
binary sequences, which we introduce, and several explicit computations are
provided. In addition, in the appendix by K. Sackel the F(q)-rational points of
its graded augmentation variety are shown to coincide with (q+1)-colorings of
the dual graph.Comment: 33 pages, 22 figure
A classification of automorphisms of compact 3-manifolds
We classify isotopy classes of automorphisms (self-homeomorphisms) of
3-manifolds satisfying the Thurston Geometrization Conjecture. The
classification is similar to the classification of automorphisms of surfaces
developed by Nielsen and Thurston, except an automorphism of a reducible
manifold must first be written as a suitable composition of two automorphisms,
each of which fits into our classification. Given an automorphism, the goal is
to show, loosely speaking, either that it is periodic, or that it can be
decomposed on a surface invariant up to isotopy, or that it has a "dynamically
nice" representative, with invariant laminations that "fill" the manifold.
We consider automorphisms of irreducible and boundary-irreducible 3-manifolds
as being already classified, though there are some exceptional manifolds for
which the automorphisms are not understood. Thus the paper is particularly
aimed at understanding automorphisms of reducible and/or boundary reducible
3-manifolds.
Previously unknown phenomena are found even in the case of connected sums of
products of a 2-sphere with a 1-sphere. To deal with this case, we prove that a
minimal genus Heegaard decomposition is unique up to isotopy, a result which
apparently was previously unknown.
Much remains to be understood about some of the automorphisms of the
classification.Comment: 61 pages, 8 eps figure
Closed Reeb orbits on the sphere and symplectically degenerate maxima
We show that the existence of one simple closed Reeb orbit of a particular
type (a symplectically degenerate maximum) forces the Reeb flow to have
infinitely many periodic orbits. We use this result to give a different proof
of a recent theorem of Cristofaro-Gardiner and Hutchings asserting that every
Reeb flow on the standard contact three-sphere has at least two periodic
orbits. Our methods are based on adapting the machinery originally developed
for proving the Hamiltonian Conley conjecture to the contact setting
Ergodic components and topological entropy in geodesic flows of surfaces
We consider the geodesic flow of reversible Finsler metrics on the 2-sphere
and the 2-torus, whose geodesic flow has vanishing topological entropy.
Following a construction of A. Katok, we discuss examples of Finsler metrics on
both surfaces, which have large ergodic components for the geodesic flow in the
unit tangent bundle. On the other hand, using results of J. Franks and M.
Handel, we prove that ergodicity and dense orbits cannot occur in the full unit
tangent bundle of the 2-sphere, if the Finsler metric has positive flag
curvatures and at least two closed geodesics. In the case of the 2-torus, we
show that ergodicity is restricted to strict subsets of tubes between
flow-invariant tori in the unit tangent bundle of the 2-torus.Comment: 23 pages, 2 figure
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