Hyperbolic manifold in the context of "Riemann surface"

In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space in which the system moves, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity.

Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motions of particles, that is, geodesics, on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the same general area, eventually filling the entire space.

In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals. Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule.