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.
A topological dynamical system consists of a compact Hausdorff topological space X and a homeomorphism
Lindenstrauss introduced the definition of orbit capacity:[1] Here,
is the membership function for the set
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
These sets occur in the definition of the small boundary property.