We study quantum scattering on manifolds equivalent to the Euclidean space near infinity, in the semiclassical regime. We assume that the corresponding classical flow admits a non-trivial trapped set, and that the dynamics on this set is of Axiom A type (uniformly hyperbolic). We are interested in the distribution of quantum resonances near the real axis. In two dimensions, we prove that, if the trapped set is sufficiently “thin”, then there exists a gap between the resonances and the real axis (that is, quantum decay rates are bounded from below). In higher dimension, the condition for this gap is given in terms of a certain topological pressure associated with the classical flow. Under the same assumption, we also prove a resolvent estimate with a logarithmic loss compared to non-trapping situations.

We prove an analog of Böttcher’s theorem for transcendental entire functions in the Eremenko–Lyubich class $ \mathcal{B} $. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are quasiconformally equivalent in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points that remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane.

We also prove that the conjugacy is essentially unique. In particular, we show that a function $ f \in \mathcal{B} $ has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic functions $ f,g \in \mathcal{B} $ that belong to the same parameter space are conjugate on their sets of escaping points.

A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the $ \mathfrak{g} ={\rm{A}}_{n}$ case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra $ \mathfrak{g} $.