In mathematics, the orbit capacity of a subset of a topological dynamical system may be thought of heuristically as a âÂÂtopological dynamical probability measureâ 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.
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.