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Resonant spaces for volume-preserving Anosov flows. (English) Zbl 1467.37031

This paper is about resonant spaces of invariant distributions with values in the bundle of exterior forms of volume-preserving Anosov flows on three-dimensional manifolds. The authors consider a closed three-manifold \(M\) with a volume form \(\Omega\) and a volume-preserving Anosov flow \(\phi_t\) having infinitesimal generator \(X\). They write the Anosov splitting as \(TM = \mathbb{R}X \oplus E_s \oplus E_u\), with \(E_0^*, E_s^*\), and \(E_u^*\) the duals of \(\mathbb{R}X, E_s\), and \( E_u\) respectively. The space of distributions with values in the bundle of exterior \(k\)-forms (and with wave front set contained in \(E_u^*\)) is denoted \(\mathcal{D}'_{E_u^*}\). The resonant spaces of interest to the authors are Res\(_k(0) = \{u \in \mathcal{D}'_{E_u^*} (M, \Omega^k) : \iota_X u =0 , \iota_X du = 0 \}\).
The authors’ first main result gives the dimensions of the resonant spaces Res\(_k(0)\) for \(k = 0,1, 2\) in terms of the first Betti number of \(M\). The dimensions depend on the cohomology class \([\omega]\) (where \(\omega = \iota_X \Omega\)) and, when \([\omega] = 0\), on the helicity \(\mathcal{H} = \int_M \tau (X) \Omega\) where \(\tau\) is any one-form with \(d\tau = \omega\).
The authors note that these dimensions agree with the Pollicott-Ruelle resonance multiplicities when either \(X\) or \(\phi_t\) is semisimple. Later in the paper they explore consequences of their results with respect to the order of vanishing at zero of the Ruelle zeta function.

MSC:

37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D10 Invariant manifold theory for dynamical systems
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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