Abstract
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 $\prec$.
Highlights
Let X be the full shift {0, 1}N on two symbols, and consider MX, the collection of shift-invariant Borel probability measures on X
If X is equipped with the lexicographic order, MX can be equipped with the partial order of first-order stochastic dominance: if μ and ν are shift-invariant Borel probability measures on X, ν dominates μ, if X f dμ ≤ X f dν for all increasing functions f : X → R
In the setting of this article, first-order stochastic dominance is used to make precise the notion of one probability measure being larger than another. The study of this order in such a setting is motivated by interesting questions that arise in ergodic optimization: the study of the smallest and largest possible ergodic averages of a given function and of the invariant measures which attain these extrema, known as minimizing and maximizing measures, respectively
Summary
Let X be the full shift {0, 1}N on two symbols, and consider MX , the collection of shift-invariant Borel probability measures on X. Proving incomparability between measures directly from the definition of dominance (or its reformulations) becomes increasingly difficult as the period of the orbits carrying the corresponding measures gets larger In this last example, one may observe that all of these measures have frequency (defined as the measure of the cylinder set 1 ) equal to 1/2.
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