Negative-curved spaces can be warped so that every single repeating path takes a whole number of seconds to complete.
Geometry in curved space is usually messy and results in irrational numbers for path lengths. This new mathematical construction creates a continuous reparameterization of flows where every periodic orbit has an integer length. It essentially turns a chaotic environment into a perfectly synchronized clockwork universe. This is the first time anyone has shown that such a structured arrangement is possible in these types of manifolds. It provides a new tool for studying the deep relationship between geometry and time-keeping.
A geometric correspondence for reparameterizations of geodesic flows
arXiv · 2605.02585
For any non-elementary, torsion-free hyperbolic group, we provide a correspondence between the left-invariant Gromov-hyperbolic metrics on the group that are quasi-isometric to a word metric, and continuous reparameterizations of the associated Mineyev's flow space. From this correspondence, we produce the first examples of continuous reparameterizations of geodesic flows on negatively curved manifolds with all periodic orbits having integer lengths. For surface and free groups, this also yields