• Source: Orbit capacity

      • 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


      A topological dynamical system consists of a compact Hausdorff topological space X and a homeomorphism



      T
      :
      X
      →
      X


      {\displaystyle T:X\rightarrow X}

      . Let



      E
      ⊂
      X


      {\displaystyle E\subset X}

      be a set. Lindenstrauss introduced the definition of orbit capacity:




      ocap
      ⁡
      (
      E
      )
      =

      lim

      n
      →
      ∞



      sup

      x
      ∈
      X




      1
      n



      ∑

      k
      =
      0


      n
      −
      1



      χ

      E


      (

      T

      k


      x
      )


      {\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)}


      Here,




      χ

      E


      (
      x
      )


      {\displaystyle \chi _{E}(x)}

      is the membership function for the set



      E


      {\displaystyle E}

      . That is




      χ

      E


      (
      x
      )
      =
      1


      {\displaystyle \chi _{E}(x)=1}

      if



      x
      ∈
      E


      {\displaystyle x\in E}

      and is zero otherwise.


      Properties


      One has



      0
      ≤
      ocap
      ⁡
      (
      E
      )
      ≤
      1


      {\displaystyle 0\leq \operatorname {ocap} (E)\leq 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:




      ocap
      ⁡
      (
      A
      ∪
      B
      )
      ≤
      ocap
      ⁡
      (
      A
      )
      +
      ocap
      ⁡
      (
      B
      )


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


      For a closed set C,




      ocap
      ⁡
      (
      C
      )
      =

      sup

      μ
      ∈

      M

      T


      ⁡
      (
      X
      )


      μ
      (
      C
      )


      {\displaystyle \operatorname {ocap} (C)=\sup _{\mu \in \operatorname {M} _{T}(X)}\mu (C)}


      Where MT(X) is the collection of T-invariant probability measures on X.


      Small sets


      When



      ocap
      ⁡
      (
      A
      )
      =
      0


      {\displaystyle \operatorname {ocap} (A)=0}

      ,



      A


      {\displaystyle A}

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


      References

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