Orbit capacity explained

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

T:X → X . Let E\subset X be a set. Lindenstrauss introduced the definition of orbit capacity:^[1] \operatorname{ocap}(E) = \lim _{n → infty} \sup _{x\in} (1)/(n) \sum ^{n -1} _{k =0} \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\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:

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

• For a closed set C,

\operatorname{ocap}(C) = \sup _{\mu\in} _{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.

Notes and References

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