 Award ID(s):
 2041823
 NSFPAR ID:
 10332895
 Date Published:
 Journal Name:
 Transactions of the American Mathematical Society
 ISSN:
 00029947
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract We show that for a generic conformal metric perturbation of a compact hyperbolic 3manifold $$\Sigma $$ Σ with Betti number $$b_1$$ b 1 , the order of vanishing of the Ruelle zeta function at zero equals $$4b_1$$ 4  b 1 , while in the hyperbolic case it is equal to $$42b_1$$ 4  2 b 1 . This is in contrast to the 2dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2forms on the sphere bundle $$S\Sigma $$ S Σ with harmonic 1forms on $$\Sigma $$ Σ .more » « less

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Abstract The question of global existence versus finitetime singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation
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