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Index problem for elliptic operators associated with a diffeomorphism of a manifold and uniformization. (English. Russian original) Zbl 1248.47046
Dokl. Math. 84, No. 3, 846-849 (2011); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 441, No. 5, 593-596 (2011).
Let \(M\) be a smooth manifold on which a smooth isometric diffeomorphism \(g: M\to M\) is given. The powers of \(g\) generate an action of the group \(\mathbb Z\) on \(M.\) The authors consider the differential operator with shift on \(M\) defined by \[ D=\sum D_kT^k: C^\infty(M)\to C^\infty(M), \] where \(T^ku(x)=u(g^k(x))\) and \(D_k\) are differential operators. The operator \(T\) is Fredholm under an appropriate ellipticity condition, and the main purpose of the note under review is to calculate its index.

47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
58J20 Index theory and related fixed-point theorems on manifolds
58B15 Fredholm structures on infinite-dimensional manifolds
Full Text: DOI
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