12,725 research outputs found
Isomorphism and embedding of Borel systems on full sets
A Borel system consists of a measurable automorphism of a standard Borel
space. We consider Borel embeddings and isomorphisms between such systems
modulo null sets, i.e. sets which have measure zero for every invariant
probability measure. For every t>0 we show that in this category there exists a
unique free Borel system (Y,S) which is strictly t-universal in the sense that
all invariant measures on Y have entropy <t, and if (X,T) is another free
system obeying the same entropy condition then X embeds into Y off a null set.
One gets a strictly t-universal system from mixing shifts of finite type of
entropy at least t by removing the periodic points and "restricting" to the
part of the system of entropy <t. As a consequence, after removing their
periodic points the systems in the following classes are completely classified
by entropy up to Borel isomorphism off null sets: mixing shifts of finite type,
mixing positive-recurrent countable state Markov chains, mixing sofic shifts,
beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular
any two equal-entropy systems from these classes are entropy conjugate in the
sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.Comment: 17 pages, v2: correction to bibliograph
Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps
We introduce "puzzles of quasi-finite type" which are the counterparts of our
subshifts of quasi-finite type (Invent. Math. 159 (2005)) in the setting of
combinatorial puzzles as defined in complex dynamics. We are able to analyze
these dynamics defined by entropy conditions rather completely, obtaining a
complete classification with respect to large entropy measures and a
description of their measures with maximum entropy and periodic orbits. These
results can in particular be applied to entropy-expanding maps like
(x,y)-->(1.8-x^2+sy,1.9-y^2+sx) for small s. We prove in particular the
meromorphy of the Artin-Mazur zeta function on a large disk. This follows from
a similar new result about strongly positively recurrent Markov shifts where
the radius of meromorphy is lower bounded by an "entropy at infinity" of the
graph.Comment: accepted by Annales de l'Institut Fourier, final revised versio
Graph immersions with parallel cubic form
We consider non-degenerate graph immersions into affine space whose cubic form is parallel with respect to the Levi-Civita
connection of the affine metric. There exists a correspondence between such
graph immersions and pairs , where is an -dimensional real
Jordan algebra and is a non-degenerate trace form on . Every graph
immersion with parallel cubic form can be extended to an affine complete
symmetric space covering the maximal connected component of zero in the set of
quasi-regular elements in the algebra . It is an improper affine hypersphere
if and only if the corresponding Jordan algebra is nilpotent. In this case it
is an affine complete, Euclidean complete graph immersion, with a polynomial as
globally defining function. We classify all such hyperspheres up to dimension
5. As a special case we describe a connection between Cayley hypersurfaces and
polynomial quotient algebras. Our algebraic approach can be used to study also
other classes of hypersurfaces with parallel cubic form.Comment: some proofs have been simplified with respect to the first versio
Toric varieties and spherical embeddings over an arbitrary field
We are interested in two classes of varieties with group action, namely toric
varieties and spherical embeddings. They are classified by combinatorial
objects, called fans in the toric setting, and colored fans in the spherical
setting. We characterize those combinatorial objects corresponding to varieties
defined over an arbitrary field . Then we provide some situations where
toric varieties over are classified by Galois-stable fans, and spherical
embeddings over by Galois-stable colored fans. Moreover, we construct an
example of a smooth toric variety under a 3-dimensional nonsplit torus over
whose fan is Galois-stable but which admits no -form. In the spherical
setting, we offer an example of a spherical homogeneous space over \mr
of rank 2 under the action of SU(2,1) and a smooth embedding of whose fan
is Galois-stable but which admits no \mr-form
Zero-dimensional symplectic isolated complete intersection singularities
We study the local symplectic algebra of the 0-dimensional isolated complete
intersection singularities. We use the method of algebraic restrictions to
classify these symplectic singularities. We show that there are non-trivial
symplectic invariants in this classification.Comment: 9 pages. arXiv admin note: text overlap with arXiv:1101.517
Boundary quotients and ideals of Toeplitz C*-algebras of Artin groups
We study the quotients of the Toeplitz C*-algebra of a quasi-lattice ordered
group (G,P), which we view as crossed products by a partial actions of G on
closed invariant subsets of a totally disconnected compact Hausdorff space, the
Nica spectrum of (G,P). Our original motivation and our main examples are drawn
from right-angled Artin groups, but many of our results are valid for more
general quasi-lattice ordered groups. We show that the Nica spectrum has a
unique minimal closed invariant subset, which we call the boundary spectrum,
and we define the boundary quotient to be the crossed product of the
corresponding restricted partial action. The main technical tools used are the
results of Exel, Laca, and Quigg on simplicity and ideal structure of partial
crossed products, which depend on amenability and topological freeness of the
partial action and its restriction to closed invariant subsets. When there
exists a generalised length function, or controlled map, defined on G and
taking values in an amenable group, we prove that the partial action is
amenable on arbitrary closed invariant subsets. Our main results are obtained
for right-angled Artin groups with trivial centre, that is, those with no
cyclic direct factor; they include a presentation of the boundary quotient in
terms of generators and relations that generalises Cuntz's presentation of O_n,
a proof that the boundary quotient is purely infinite and simple, and a
parametrisation of the ideals of the Toeplitz C*-algebra in terms of subsets of
the standard generators of the Artin group.Comment: 26 page
Symplectic , singularities and Lagrangian tangency orders
We study the local symplectic algebra of curves. We use the method of
algebraic restrictions to classify symplectic singularities. We define
discrete symplectic invariants - the Lagrangian tangency orders. We use these
invariants to distinguish symplectic singularities of classical
singularities of planar curves, singularity and singularity. We
also give the geometric description of these symplectic singularities
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