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

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A topological dynamical system consists of a compact Hausdorff topological space X and a homeomorphism  . Let   be a set. Lindenstrauss introduced the definition of orbit capacity:[1]

 

Here,   is the membership function for the set  . That is   if   and is zero otherwise.

Properties

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One has  . 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:

 
  • For a closed set C,
 
Where MT(X) is the collection of T-invariant probability measures on X.

Small sets

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When  ,   is called small. These sets occur in the definition of the small boundary property.

References

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  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.