This talk will try to remain at an elementary technical level while its main purpose is to build a bridge between hard-core people working on spectral theory of (non)self-adjoint operators, and hard-core people working on discrete dynamical systems. To be more precise, for $\beta>1$ we consider a discrete dynamical system induced by the map $T:[0,1)\to [0,1)$ where $T(x)=\beta x- \lfloor \beta x\rfloor$ and we investigate some spectral and dynamical (ergodic) properties of its associated (non-selfadjoint) Perron-Frobenius operator. Also, when $\beta\geq 2$ is an integer, we establish an unexpected connection to the classical Euler-Bernoulli approximation formula. The talk is based on joint works with G. Marcelli and I. Herbst (to appear in J. Spec. Th. https://arxiv.org/abs/2502.06511) and with K. Sørensen (https://www.sciencedirect.com/science/article/pii/S0021904525000929).