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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle E\subset X} be a set. Lindenstrauss introduced the definition of orbit capacity:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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, $\chi _{E}(x)$ is the membership function for the set $E$. That is $\chi _{E}(x)=1$ if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in E} and is zero otherwise.

Properties

One has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{ocap}(A\cup B)\leq \operatorname{ocap}(A)+\operatorname{ocap}(B)}

• For a closed set C,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{ocap}(A)=0} , $A$ is called small. These sets occur in the definition of the small boundary property.