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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.

MSC:
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
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