# constant averages along orbits

What should one say to describe the situation in which a function $T$ from some set $X$ to itself, and a function $f$ from $X$ to some characteristic-zero field $K$, have the property that the average of $f$ over a $T$-orbit in $X$ is the same for each orbit? (I'm mostly concerned with the case in which every orbit is finite, so that the notion of average is the naive one.)

For about a year, in collaboration with Tom Roby and others, I've been studying such situations, which turn out to crop up everywhere in combinatorics. I think that the topic needs to have some appropriate (and not too unwieldy) terminology associated with it, but nothing seems to exist in the literature, so I'm left with the choice of adopting a parallel notion from an allied field or coining something new. But I don't love anything I've come up with so far.

(See functions whose average along orbits is zero or a constant for an earlier post of mine on this topic.)

Here are some terms I've considered using to denote "Property X" (with explanatory notes following the list):

#1. "$T$ is pseudotransitive relative to $f$"

#2. "The triple $(X,T,f)$ has the CAAO (Constant Averages Along Orbits) Property"

#3. "$f$ is CPC (Constant Plus Coboundary) relative to $T$"

#4. "The triple $(X,T,f)$ exhibits combinatorial ergodicity"

#5. "$f$ is convariant [sic] under $T$"

#6. "$f$ is mixed by $T$", "$T$ mixes $f$"

#7. "$f$ is balanced with respect to $T$"

#8. "$f$ is centered relative to $T$"

#9. "$f$ is Cesaro-constant under the action of $T$"

#10. "$f$ is $T$-constant"

#11. "$T$ and $f$ are disjoint"

I'm hoping a community wiki discussion might help me settle on some good nomenclature (or at least point me toward analogues of what I'm looking at in other fields of mathematics).

Notes:

#1. "$T$ is pseudotransitive relative to $f$"

If $X$ consists of a single $T$-orbit, then Property X holds trivially. So one might paraphrase Property X as: "The action is behaving like a transitive action even though it isn't (necessarily) one." But one problem with saying that $T$ is pseudotransitive relative to $f$ is that is doesn't suggest a companion nomenclature for what $f$ is relative to $T$ (which is important in my research). One can't say "$f$ is pseudotransitivized by $T$"!

#2. "The triple $(X,T,f)$ has the CAAO (Constant Averages Along Orbits) Property"

This has the virtue of being quite descriptive. And I think I could write both "$T$ is CAAO relative to $f$" and "$f$ is CAAO relative to $T$" without embarrassment. Moreover, CAAO works well as an acronym; that is, unlike CPC, which works only as an initialism, CAAO can be pronounced ("cow"). But the initially amusing homophony may not age well. (Whimsy can wear thin after a few decades.)

#3. "$f$ is CPC (Constant Plus Coboundary) relative to $f$"

A function $f$ has Property X relative to $T: X \rightarrow X$ if and only $f$ can be written as $f(x) = c + g(x) - g(T(x))$, where $c$ is some constant and $g$ is some (non-unique) function from $X$ to $K$. (One can also say that $f$ is cohomologous to a constant.) I'd be happier with the constant-plus-coboundary nomenclature if the $g$-functions turned out to play an important role in examples, which so far hasn't been the case. Also, I fear that the phrase "CPC phenomenon" would invite confusion with the phrases "CSP" and "cyclic sieving phenomenon", which frequently arises in the same combinatorial situations as Property X.

#4. "The triple $(X,T,f)$ exhibits combinatorial ergodicity"

I've used this one in my talks. I like it, since Boltzmann's notion of ergodicity is precisely that long-term averages are the same for all orbits (and if all orbits are finite, long-term averages are the same as orbit averages). People have sometimes objected that in dynamics and in physics, ergodicity is something that pertains to a mapping $T$, not a mapping $T$ relative to a function $f$. I've replied that the word ergodic always means relative to a set of functions (measurable functions if one is doing ergodic theory, macroscopic functions if one is doing physics), even if that relativity is left implicit. So why not make that relationship explicit, and say that what's really ergodic is a map $T$ with respect to a function or set of functions? I was happy with this for a while. But one can't say "$f$ is ergodic relative to $T$"; that stretches the metaphor too far for my taste. And I need a crisp way of referring to the functions $f$ such that $(X,T,f)$ has Property X, for some particular map $T: X \rightarrow X$.

#5. "$f$ is convariant under $T$"

Yes, I mean convariant, not covariant, which already means something else. "Convariant" is meant to be a counterpart to "invariant", since every function from $X$ to $K$ can be written as the sum of an invariant function (that is, a function $h$ satisfying $h(T(x))=h(x)$ for all $x$) and a function with Property X. Note that the invariant functions form a subspace, as do the convariant functions. So from a linear algebra perspective, it's a nice situation.

#6. "$f$ is mixed by $T$", "$T$ mixes $f$"

I like this in part because of the underlying physical intuition (we say a solution of 90% water and 10% salt had been mixed if every portion of the solution has water and salt in those same proportions; replace "portion" by "orbit" and you're fairly close to Property X). Also, one can refer to the "invariant" and "convariant" functions as being respectively "fixed" and "mixed" by $T$, which is cute (but not too cute!) and certainly succinct. Yet I worry that the ergodic theory meaning of the word "mixing", which carries connotations stronger than ergodicity, may be distracting or even confusing for some people.

#7. "$f$ is balanced with respect to $T$"

This is on the bland and vague side, but I can't completely dismiss it.

#8. "$f$ is centered relative to $T$"

Ditto.

#9. "$f$ is Cesaro-constant under the action of $T$"

Property X says that if we define $F(x)$ as the Cesaro mean of $f(x),f(T(x)),f(T(T(x))),...$, then $F$ is constant over $X$ (where the Cesaro mean of a sequence is the limit as $n$ goes to infinity of the mean of the first $n$ terms).

#10. "$f$ is $T$-constant"

This is intended as a shorthand for "$f$ is constant modulo $T$-coboundaries".

#11. "$T$ and $f$ are disjoint"

I don't know a sense (coming from some allied field) in which the word "disjoint" might be applicable to Property X, but I suspect that there might be.

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Regarding #11: Disjointness already has an established meaning in ergodic theory: ams.org/mathscinet-getitem?mr=213508 . It is not the same usage as the one given here, so I would not recommend this usage. Personally, I would go with something straightforward, such as "f has constant T-averages". – Terry Tao Sep 30 '12 at 23:54
$f$ is a constant plus a coboundary? – Anthony Quas Oct 1 '12 at 20:28
@ Anthony Quas: Yes; see #3 in my post. – James Propp Oct 3 '12 at 4:48
@Terry Tao: I think there might be a connection between Property X and the notion of disjointness in ergodic theory for maps with discrete spectrum, though I haven't been able to find a satisfactory link. E.g., let $X = R^n/Z^n$, $T(x) = x+u$ for some fixed $u$, and $f(x) = \exp(2 \pi i x \cdot v)$ for some fixed $v$. In this situation, or something like it, I think $(X,T,f)$ has Property X iff the spectra associated with $T$ and $f$ are disjoint (though I don't know what "the spectrum associated with $f$" might actually mean). – James Propp Oct 3 '12 at 4:53
@James: I don't know if you still care about mathoverflow.net/questions/62340/…, but I saw a problem with your proofs of 17 and 18. The claim "every non-empty bounded interval has endpoints" is itself equivalent to Dedekind completeness, but you only show 17 and 18 for closed intervals with endpoints. – Ricky Demer Oct 18 '12 at 21:22

I've decided to go with the terms "homomesy" and "homomesic" (from Greek roots meaning "same middle"), suggested by my collaborator Tom Roby.

To see why I think the the concept deserves a name, check out the examples given in the slide-presentation http://jamespropp.org/mitcomb13a.pdf (which barely scratches the surface of all the examples of the phenomenon that have come to light over the past year).

To see a context in which the concept of homomesy is precisely dual to the concept of invariance, see http://jamespropp.org/Dec2012a.pdf .

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