Orbit capacity

In mathematics, the orbit capacity of a subset of a topological [dynamical system] may be thought of heuristically as a “topological dynamical probability measure” of the subset. More precisely, its value for a set is a tight upper bound for the normalized number of visits of orbits in this set.

Definition

A topological dynamical system consists of a compact Hausdorff topological space X and a homeomorphism. Let be a set. Lindenstrauss introduced the definition of orbit capacity:
Here, is the membership function for the set. That is if and is zero otherwise.

One has. By convention, topological dynamical systems do not come equipped with a measure; the orbit capacity can be thought of as defining one, in a "natural" way. It is not a true measure, it is only a sub-additive:

When, is called small. These sets occur in the definition of the small boundary property.