429 research outputs found
Twisted K-theory of differentiable stacks
In this paper, we develop twisted -theory for stacks, where the twisted
class is given by an -gerbe over the stack. General properties, including
the Mayer-Vietoris property, Bott periodicity, and the product structure
are derived. Our
approach provides a uniform framework for studying various twisted -theories
including the usual twisted -theory of topological spaces, twisted
equivariant -theory, and the twisted -theory of orbifolds. We also
present a Fredholm picture, and discuss the conditions under which twisted
-groups can be expressed by so-called "twisted vector bundles".
Our approach is to work on presentations of stacks, namely \emph{groupoids},
and relies heavily on the machinery of -theory (-theory) of
-algebras.Comment: 74 page
A temporal Central Limit Theorem for real-valued cocycles over rotations
We consider deterministic random walks on the real line driven by irrational
rotations, or equivalently, skew product extensions of a rotation by
where the skewing cocycle is a piecewise constant mean zero function with a
jump by one at a point . When is badly approximable and
is badly approximable with respect to , we prove a Temporal Central
Limit theorem (in the terminology recently introduced by D.Dolgopyat and
O.Sarig), namely we show that for any fixed initial point, the occupancy random
variables, suitably rescaled, converge to a Gaussian random variable. This
result generalizes and extends a theorem by J. Beck for the special case when
is quadratic irrational, is rational and the initial point is
the origin, recently reproved and then generalized to cover any initial point
using geometric renormalization arguments by Avila-Dolgopyat-Duryev-Sarig
(Israel J., 2015) and Dolgopyat-Sarig (J. Stat. Physics, 2016). We also use
renormalization, but in order to treat irrational values of , instead of
geometric arguments, we use the renormalization associated to the continued
fraction algorithm and dynamical Ostrowski expansions. This yields a suitable
symbolic coding framework which allows us to reduce the main result to a CLT
for non homogeneous Markov chains.Comment: a few typos corrected, 28 pages, 4 figure
Isomorphism of uniform algebras on the 2-torus
For \alpha a positive irrational, we consider the uniform subalgebra A_\alpha of C(T^2) consisting of those functions f satisfying \hat{f}(m,n)=0 whenever m+n\alpha\u3c0. For positive irrationals \alpha, \beta, we determine when A_\alpha and A_\beta are isometrically isomorphic. Furthermore, we describe the group Aut(A_\alpha) of isometric automorphisms of A_\alpha. Finally we show how an explicit representation of Aut(A_\alpha) can be derived from Pell\u27s equations
A strictly stationary -mixing process satisfying the central limit theorem but not the weak invariance principle
In 1983, N. Herrndorf proved that for a -mixing sequence satisfying the
central limit theorem and , the weak
invariance principle takes place. The question whether for strictly stationary
sequences with finite second moments and a weaker type (, ,
) of mixing the central limit theorem implies the weak invariance
principle remained open.
We construct a strictly stationary -mixing sequence with finite
moments of any order and linear variance for which the central limit theorem
takes place but not the weak invariance principle.Comment: 12 page
The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps
A pair , where is a locally
rectifiable curve and is a {\em Heisenberg
uniqueness pair} if an absolutely continuous (with respect to arc length)
finite complex-valued Borel measure supported on whose Fourier
transform vanishes on necessarily is the zero measure. Recently, it
was shown by Hedenmalm and Montes that if is the hyperbola
, where is the mass, and is the
lattice-cross , where are positive reals, then
is a Heisenberg uniqueness pair if and only if . The Fourier transform of a measure supported on a hyperbola
solves the one-dimensional Klein-Gordon equation, so the theorem supplies very
thin uniqueness sets for a class of solutions to this equation. The case of the
semi-axis as well as the holomorphic counterpart remained open.
In this work, we completely solve these two problems. As for the semi-axis, we
show that the restriction to of the above exponential system
spans a weak-star dense subspace of if and only if
, based on dynamics of Gauss-type maps. This has an
interpretation in terms of dynamical unique continuation. As for the
holomorphic counterpart, we show that the above exponential system with
spans a weak-star dense subspace of if and
only if . To obtain this result, we need to develop new
harmonic analysis tools for the dynamics of Gauss-type maps, related to the
Hilbert transform. Some details are deferred to a separate publication.Comment: 41 page
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