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 $0\leq \operatorname {ocap} (E)\leq 1$. 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 M_T(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.