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 $$T:X\rightarrow X$$. Let $$E\subset X$$ be a set. Lindenstrauss introduced the definition of orbit capacity:


 * $$\operatorname{ocap}(E)=\lim_{n\rightarrow\infty}\sup_{x\in X} \frac 1 n \sum_{k=0}^{n-1} \chi_E (T^k x)$$

Here, $$\chi_E(x)$$ is the membership function for the set $$E$$. That is $$\chi_E(x)=1$$ if $$x\in E$$ and is zero otherwise.

Properties
One has $$0\le\operatorname{ocap}(E)\le 1$$. 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:


 * Orbit capacity is sub-additive:


 * $$\operatorname{ocap}(A\cup B)\leq \operatorname{ocap}(A)+\operatorname{ocap}(B)$$


 * For a closed set C,


 * $$\operatorname{ocap}(C)=\sup_{\mu\in \operatorname{M}_T(X)}\mu(C)$$


 * Where MT(X) is the collection of T-invariant probability measures on X.

Small sets
When $$\operatorname{ocap}(A)=0$$, $$A$$ is called small. These sets occur in the definition of the small boundary property.