  The black hole photon ring is a prime target for upcoming space-based VLBI
missions seeking to image the fine structure of astrophysical black holes. The
classical Lyapunov exponents of the corresponding nearly bound null geodesics
control the quasinormal ringing of a perturbed black hole as it settles back
down to equilibrium, and they admit a holographic interpretation in terms of
quantum Ruelle resonances of the microstate dual to the Kerr black hole. Recent
work has identified a number of emergent symmetries related to the intricate
self-similar structure of the photon ring. Here, we explore this web of
interrelated phenomena in an exactly soluble example that arises as an
approximation to the near-extremal Kerr black hole. The self-dual warped
AdS$_3$ geometry has a photon ring as well as $\mathsf{SL}(2,\mathbb{R})$
isometries and an exactly calculable quasinormal mode (QNM) spectrum. We show
explicitly that the geometric optics approximation reproduces the eikonal limit
of the exact QNM spectrum, as well as the approximate "near-ring"
wavefunctions. The $\mathsf{SL}(2,\mathbb{R})$ isometries are directly related
to the emergent conformal symmetry of the photon ring in black hole images but
are distinct from a recently discussed conformal symmetry of the eikonal QNM
spectrum. The equivalence of the classical QNM spectrum -- and thus the photon
ring -- to the quantum Ruelle resonances in the context of a spacetime with a
putative holographic dual suggests that the photon ring of a warped black hole
is indeed part of the black hole hologram.
