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Let I be an open interval containing zero, let M be a real manifold, let P T M be its cotangent bundle with the zero-section removed, and letˆD ¹' t º t 2I be a homogeneous Hamiltonian isotopy of

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We interpret some results of persistent homology and barcodes (in any dimension) with the language of microlocal sheaf theory. For that purpose we study the derived category of sheaves on a real finite-dimensional vector space V. By using the operation of convolution, we introduce a pseudo-distance on this category and prove in particular a stability result for direct images. Then we assume that V is endowed with a closed convex proper cone γ with non empty interior and study γ-sheaves, that is, constructible sheaves with microsupport contained in the antipodal to the polar cone (equivalently, constructible sheaves for the γ-topology).We prove that such sheaves may be approximated (for the pseudo-distance) by "piecewise linear" γ-sheaves. Finally we show that these last sheaves are constant on stratifications by γ-locally closed sets, an analogue of barcodes in higher dimension.

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