110 research outputs found

    Slow Diffeomorphisms of a Manifold with Two Dimensions Torus Action

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    The uniform norm of the differential of the n-th iteration of a diffeomorphism is called the growth sequence of the diffeomorphism. In this paper we show that there is no lower universal growth bound for volume preserving diffeomorphisms on manifolds with an effective two dimensions torus action by constructing a set of volume-preserving diffeomorphisms with arbitrarily slow growth.Comment: 12 p

    Kick stability in groups and dynamical systems

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    We consider a general construction of ``kicked systems''. Let G be a group of measure preserving transformations of a probability space. Given its one-parameter/cyclic subgroup (the flow), and any sequence of elements (the kicks) we define the kicked dynamics on the space by alternately flowing with given period, then applying a kick. Our main finding is the following stability phenomenon: the kicked system often inherits recurrence properties of the original flow. We present three main examples. 1) G is the torus. We show that for generic linear flows, and any sequence of kicks, the trajectories of the kicked system are uniformly distributed for almost all periods. 2) G is a discrete subgroup of PSL(2,R) acting on the unit tangent bundle of a Riemann surface. The flow is generated by a single element of G, and we take any bounded sequence of elements of G as our kicks. We prove that the kicked system is mixing for all sufficiently large periods if and only if the generator is of infinite order and is not conjugate to its inverse in G. 3) G is the group of Hamiltonian diffeomorphisms of a closed symplectic manifold. We assume that the flow is rapidly growing in the sense of Hofer's norm, and the kicks are bounded. We prove that for a positive proportion of the periods the kicked system inherits a kind of energy conservation law and is thus superrecurrent. We use tools of geometric group theory and symplectic topology.Comment: Latex, 40 pages, revised versio

    A two-cocycle on the group of symplectic diffeomorphisms

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    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

    Quelques plats pour la m\'etrique de Hofer

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    We show, by an elementary and explicit construction, that the group of Hamiltonian diffeomorphisms of certain symplectic manifolds, endowed with Hofer's metric, contains subgroups quasi-isometric to Euclidean spaces of arbitrary dimension.Comment: 9 pages, minor change

    Continuous Wavelets on Compact Manifolds

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    Let M\bf M be a smooth compact oriented Riemannian manifold, and let ΔM\Delta_{\bf M} be the Laplace-Beltrami operator on M{\bf M}. Say 0 \neq f \in \mathcal{S}(\RR^+), and that f(0)=0f(0) = 0. For t>0t > 0, let Kt(x,y)K_t(x,y) denote the kernel of f(t2ΔM)f(t^2 \Delta_{\bf M}). We show that KtK_t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f(t2Δ)f(t^2\Delta) on \RR^n. We define continuous S{\cal S}-wavelets on M{\bf M}, in such a manner that Kt(x,y)K_t(x,y) satisfies this definition, because of its localization near the diagonal. Continuous S{\cal S}-wavelets on M{\bf M} are analogous to continuous wavelets on \RR^n in \mathcal{S}(\RR^n). In particular, we are able to characterize the Ho¨\ddot{o}lder continuous functions on M{\bf M} by the size of their continuous S{\mathcal{S}}-wavelet transforms, for Ho¨\ddot{o}lder exponents strictly between 0 and 1. If M\bf M is the torus \TT^2 or the sphere S2S^2, and f(s)=sesf(s)=se^{-s} (the ``Mexican hat'' situation), we obtain two explicit approximate formulas for KtK_t, one to be used when tt is large, and one to be used when tt is small

    Homology class of a Lagrangian Klein bottle

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    It is shown that an embedded Lagrangian Klein bottle represents a non-trivial mod 2 homology class in a compact symplectic four-manifold (X,ω)(X,\omega) with c1(X)[ω]>0c_1(X)\cdot[\omega]>0. (In versions 1 and 2, the last assumption was missing. A counterexample to this general claim and the first proof of the corrected result have been found by Vsevolod Shevchishin.) As a corollary one obtains that the Klein bottle does not admit a Lagrangian embedding into the standard symplectic four-space.Comment: Version 3 - completely rewritten to correct a mistake; Version 4 - minor edits, added references; AMSLaTeX, 6 page

    On the Use of Minimum Volume Ellipsoids and Symplectic Capacities for Studying Classical Uncertainties for Joint Position-Momentum Measurements

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    We study the minimum volume ellipsoid estimator associates to a cloud of points in phase space. Using as a natural measure of uncertainty the symplectic capacity of the covariance ellipsoid we find that classical uncertainties obey relations similar to those found in non-standard quantum mechanics

    On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions

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    Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains νn\nu_n of the nn-th eigenfunction satisfies nνnn\ge \nu_n. Here, we provide a new interpretation for the Courant nodal deficiency dn=nνnd_n = n-\nu_n in the case of quantum graphs. It equals the Morse index --- at a critical point --- of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning --- it is the number of unstable directions in the vicinity of the critical point corresponding to the nn-th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.Comment: 22 pages, 6 figure

    Symplectic geometry of quantum noise

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    We discuss a quantum counterpart, in the sense of the Berezin-Toeplitz quantization, of certain constraints on Poisson brackets coming from "hard" symplectic geometry. It turns out that they can be interpreted in terms of the quantum noise of observables and their joint measurements in operational quantum mechanics. Our findings include various geometric mechanisms of quantum noise production and a noise-localization uncertainty relation. The methods involve Floer theory and Poisson bracket invariants originated in function theory on symplectic manifolds.Comment: Revised version, 57 pages, 3 figures. Incorporates arXiv:1203.234
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