Orbit capacity

In mathematics, 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]:

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

Properties

• 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.

References

1. Lindenstrauss, Elon (1999-12-01). "Mean dimension, small entropy factors and an embedding theorem". Publications Mathématiques de l'Institut des Hautes Études Scientifiques. 89 (1): 232. doi:10.1007/BF02698858. ISSN 0073-8301.