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Stochastic dominance for shift-invariant measures

The author was partially supported by EPSRC grant EP/L02246X/1.

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  • Let $X$ be the full shift on two symbols. The lexicographic order induces a partial order known as first-order stochastic dominance on the collection ${\mathcal{M}}_{X}$ of its shift-invariant probability measures. We present a study of the fine structure of this dominance order, denoted by $\prec$, and give criteria for establishing comparability or incomparability between measures in ${\mathcal{M}}_{X}$. The criteria also give an insight to the complicated combinatorics of orbits in the shift. As a by-product, we give a direct proof that Sturmian measures are totally ordered with respect to $\prec$.

    Mathematics Subject Classification: Primary: 37B10, 37A05, 37E05, 37E15; Secondary: 37E45.


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  • Figure 1.  Orbits supporting measures $\mu_{110010} \prec \mu_{110110010} = \mu_{110}\ast\mu_{110010}\prec \mu_{110}$

    Figure 3.  A concatenation procedure that generates the largest point in the support of a Sturmian measure $S_{p/q}$

    Figure 2.  The pairwise incomparable shift-invariant probability measures $\mu_{10}$, $\mu_{1100}$, $\mu_{110100}$ and $\mu_{110010}$ of frequency $1/2$.

    Figure 4.  Hasse diagram of first-order stochastic dominance for measures supported on periodic orbits of period up to $7$. Orbits that carry measures with equal frequency are displayed on the same horizontal line, and frequencies decrease from top to bottom.

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