Orbit capacity

In mathematics, 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 formula_1. Let formula_2 be a set. Lindenstrauss introduced the definition of orbit capacity:
Here, formula_4 is the membership function for the set formula_5. That is formula_6 if formula_7 and is zero otherwise. 
Properties.
Obviously, one has formula_8. 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 sub-additive:
Small sets.
When formula_11, formula_12 is called small. These sets occur in the definition of the small boundary property.