In this talk, we will see a few approaches to obtain surfaces with either hyperbolic structures or singular Euclidean structures from 1-dimensional dynamical systems. A part of the project is related to the Fried's conjectural characterization of pseudo-Anosov stretching factors. If time permits, we will also briefly mention a statistical answer to an easy version of Fried's conjecture. This talk is partially based on joint work with J. Alonso-E. Samperton and A. Rafiqi-C. Wu.