No abstract

Consider a quantum cat map M associated to a matrix A ∈ Sp(2n, Z), which is a common toy model in quantum chaos. We show that the mass of eigenfunctions of M on any nonempty open set in the position-frequency space satisfies a lower bound which is uniform in the semiclassical limit, under two assumptions:(1) there is a unique simple eigenvalue of A of largest absolute value and (2) the characteristic polynomial of A is irreducible over the rationals. This is similar to previous work [DJ18, DJN19] on negatively curved surfaces and [Sch21] on quantum cat maps with n = 1, but this paper gives the first results of this type which apply in any dimension. When condition (2) fails we provide a weaker version of the result and discuss relations to existing counterexamples. We also obtain corresponding statements regarding semiclassical measures and damped quantum cat maps.

We define and study local and global trace formulae for discrete-time uniformly hyperbolic weighted dynamics. We explain first why dynamical determinants are particularly convenient tools to tackle this question. Then we construct counter-examples that highlight that the situation is much less well-behaved for smooth dynamics than for real-analytic ones. This suggests to study this question for Gevrey dynamics. We do so by constructing an anisotropic space of ultradistributions on which a transfer operator acts as a trace class operator. From this construction, we deduce trace formulae for Gevrey dynamics, as well as bounds on the growth of their dynamical determinants and the asymptotics of their Ruelle resonances.2 2 In fact, the main point is that we can work with a space that does not contain C ∞ (V ). 5

For analytic negatively curved Riemannian manifold with analytic strictly convex boundary, we show that the scattering map for the geodesic flow determines the manifold up to isometry. In particular one recovers both the topology and the metric. More generally our result holds in the analytic category under the no conjugate point and hyperbolic trapped sets assumptions.

Ruelle (1997 Commun. Math. Phys. 187 227–41; 2003 Commun. Math. Phys. 234 185–90) (see also Jiang 2012 Ergod. Theor. Dynam. Syst. 32 1350–69) gave a formula for linear response of transitive Anosov diffeomorphisms. Recently, practically computable realizations of Ruelle’s formula have emerged that potentially enable sensitivity analysis of certain high-dimensional chaotic numerical simulations encountered in the applied sciences. In this paper, we provide full mathematical justification for the convergence of one such efficient computation, the space–split sensitivity, or S3, algorithm (Chandramoorthy and Wang 2022 SIAM J. Appl. Dyn. Syst. 21 735–81). In S3, Ruelle’s formula is computed as a sum of two terms obtained by decomposing the perturbation vector field into a coboundary and a remainder that is parallel to the unstable direction. Such a decomposition results in a splitting of Ruelle’s formula that is amenable to efficient computation. We prove the existence of the S3 decomposition and the convergence of the computations of both resulting components of Ruelle’s formula.