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

$\mathrm{ocap}(E)=\underset{n\to \mathrm{\infty }}{\lim }\underset{x\in X}{\sup }\frac{1}{n}\sum _{k=0}^{n-1}{\chi }_E(T^{k}x)$

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 $0\le \mathrm{ocap}(E)\le 1$. 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:

$\mathrm{ocap}(A\cup B)\le \mathrm{ocap}(A)+\mathrm{ocap}(B)$

• For a closed set C,

$\mathrm{ocap}(C)=\underset{\mu \in {\mathrm{M}}_T(X)}{\sup }\mu (C)$

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

Small sets

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