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 ${\displaystyle T:X\rightarrow X}$. Let ${\displaystyle E\subset X}$ be a set. Lindenstrauss introduced the definition of orbit capacity:^[1]

${\displaystyle \operatorname {ocap} (E)=\lim _{n\rightarrow \infty }\sup _{x\in X}{\frac {1}{n}}\sum _{k=0}^{n-1}\chi _{E}(T^{k}x)}$

Here, ${\displaystyle \chi _{E}(x)}$ is the membership function for the set ${\displaystyle E}$. That is ${\displaystyle \chi _{E}(x)=1}$ if ${\displaystyle x\in E}$ and is zero otherwise.

Properties

One has ${\displaystyle 0\leq \operatorname {ocap} (E)\leq 1}$. 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:

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

• For a closed set C,

${\displaystyle \operatorname {ocap} (C)=\sup _{\mu \in \operatorname {M} _{T}(X)}\mu (C)}$

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

Small sets

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