429 research outputs found

    Twisted K-theory of differentiable stacks

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    In this paper, we develop twisted KK-theory for stacks, where the twisted class is given by an S1S^1-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure KαiKβjKα+βi+jK^i_\alpha \otimes K^j_\beta \to K^{i+j}_{\alpha +\beta} are derived. Our approach provides a uniform framework for studying various twisted KK-theories including the usual twisted KK-theory of topological spaces, twisted equivariant KK-theory, and the twisted KK-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted KK-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 KK-theory (KKKK-theory) of CC^*-algebras.Comment: 74 page

    A temporal Central Limit Theorem for real-valued cocycles over rotations

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    We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by α\alpha where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point β\beta. When α\alpha is badly approximable and β\beta is badly approximable with respect to α\alpha, 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 α\alpha is quadratic irrational, β\beta 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 β\beta, 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

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    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 β\beta-mixing process satisfying the central limit theorem but not the weak invariance principle

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    In 1983, N. Herrndorf proved that for a ϕ\phi-mixing sequence satisfying the central limit theorem and lim infnσn2n>0\liminf_{n\to\infty}\frac{\sigma^2_n}n>0, the weak invariance principle takes place. The question whether for strictly stationary sequences with finite second moments and a weaker type (α\alpha, β\beta, ρ\rho) of mixing the central limit theorem implies the weak invariance principle remained open. We construct a strictly stationary β\beta-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

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    A pair (Γ,Λ)(\Gamma,\Lambda), where ΓR2\Gamma\subset\mathbb{R}^2 is a locally rectifiable curve and ΛR2\Lambda\subset\mathbb{R}^2 is a {\em Heisenberg uniqueness pair} if an absolutely continuous (with respect to arc length) finite complex-valued Borel measure supported on Γ\Gamma whose Fourier transform vanishes on Λ\Lambda necessarily is the zero measure. Recently, it was shown by Hedenmalm and Montes that if Γ\Gamma is the hyperbola x1x2=M2/(4π2)x_1x_2=M^2/(4\pi^2), where M>0M>0 is the mass, and Λ\Lambda is the lattice-cross (αZ×{0})({0}×βZ)(\alpha\mathbb{Z}\times\{0\}) \cup (\{0\}\times\beta\mathbb{Z}), where α,β\alpha,\beta are positive reals, then (Γ,Λ)(\Gamma,\Lambda) is a Heisenberg uniqueness pair if and only if αβM24π2\alpha\beta M^2\le4\pi^2. 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 R+\mathbb{R}_+ 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 R+\mathbb{R}_+ of the above exponential system spans a weak-star dense subspace of L(R+)L^\infty(\mathbb{R}_+) if and only if 0<αβ<40<\alpha\beta<4, 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 m,n0m,n\ge0 spans a weak-star dense subspace of H+(R)H^\infty_+(\mathbb{R}) if and only if 0<αβ10<\alpha\beta\le1. 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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