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 . 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
Obviously, 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 sub-additive:
- Orbit capacity is sub-additive:
- For a closed set C,
- where MT(X) is the collection of T-invariant probability measures on X.
Small sets
When , is called small. These sets occur in the definition of the small boundary property.
References
- ↑ 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.