Near-horizon geodesic instabilities and anomalous decay of quasinormal modes in Weyl black holes

We study the stability of the Weyl geometry considering an exact black hole solution. By calculating the geodesics of massless and massive scalar fields orbiting outside the Weyl black hole background and using the Lyapunov exponent, we show that geodesic instabilities, characterized by the Lyapunov exponent, appear in the asymptotically de Sitter-like spacetime. Calculating the photon sphere's quasinormal modes (QNMs) of a scalar field perturbing the Weyl black hole, we find a relation connecting the QNMs with the Lyapunov exponent in the asymptotically de Sitter-like spacetime. Furthermore, we study the anomalous decay rate of the QNMs connecting their behavior with the Lyapunov exponent.