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 . Let be a set. Lindenstrauss introduced the definition of orbit capacity[1]:

Properties

  • 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.

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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.
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