Pingree open problems — Petersen tail-field problem 2

Definitions and mathematical statement

The statement refers to definitions in the companion problem. The local companion registry record, permanent ID 4400003, supplies $A=\{0,1,\ldots,d-1\}$ and $X=A^{{\mathbb{Z}}}$, with coordinates $x_j$, and the symbol-count vectors \[ (v_n(x))_i=\#\{0\leq j\leq n:x_j=i\},\qquad (w_n(x))_i=\#\{0\leq j\leq n:x_{-j}=i\} \] for $0\leq i<d$ and $n\geq0$. Let $\mathcal B(v_n,v_{n+1},\ldots)$ mean the sigma-field generated by those maps. The resulting forward and backward count-tail fields are \[ \mathcal F^+=\bigcap_{n\geq0}\mathcal B(v_n,v_{n+1},\ldots),\qquad \mathcal F^-=\bigcap_{n\geq0}\mathcal B(w_n,w_{n+1},\ldots). \] The companion's intermediate definition includes an intersection over an unused index $k\geq n$; as printed it repeats the same sigma-field, yielding the displayed final intersections. If $k$ rather than $n$ was intended inside the generated field, the final outer tail intersection is again the same.

For a probability measure $\mu$ on $X$, call a sigma-field $\mathcal F$ $\mu$-trivial if $\mu(E)\in\{0,1\}$ for every $E\in\mathcal F$.

Question (Karl Petersen). If the forward and backward count-tail fields do not coincide, is it nevertheless true that \[ \mathcal F^+\text{ is $\mu$-trivial} \quad\Longleftrightarrow\quad \mathcal F^-\text{ is $\mu$-trivial}? \] The record supplies no measure $\mu$ or quantifier over measures, so a complete statement must recover the intended class, for example whether shift invariance is assumed. That is not silently added. It is also unspecified whether equality/inequality of sigma-fields is literal or modulo $\mu$-null sets, and whether completions are taken. These distinctions matter for the conditional hypothesis even though triviality itself has the displayed precise meaning.

The supplied summary calls this the companion triviality problem to AMR-043-0003, says the 2010 source presents it as open, and reports no later authoritative resolution found through its 2026 search. Those are supplied evidence limitations, not independently checked current status.

Short English statement

If forward and backward tail fields of cumulative symbol counts differ, must they still be trivial together or nontrivial together? The count definitions are recovered from the local companion record. The intended probability measure, stationarity assumptions, and equality modulo null sets remain unspecified.

Sources

• Local UnsolvedMath registry, permanent ID 4400004, source code catalogue entry, "Pingree open problems --- Petersen tail-field problem 2"; proposer: Karl Petersen, proposed year 2010. Status open, classification OPEN-TRIAGE, updated 2026-08-06.
• Local companion record in data/unsolvedmath/problems.json, permanent ID 4400003, source code AMR-043-0003, "Petersen tail-field problem 1," consulted only to recover the symbol-count definitions explicitly referenced by this record. It attributes those definitions to Petersen's first problem in the same source.
• Open problems from the 3rd Pingree Workshop on Dynamical Systems (2010), Karl Petersen's second Problem, https://math.huji.ac.il/~mhochman/open-problems/pingree-open-problems.pdf. The record reports PDF-text extraction and access on 2026-07-29; no later authoritative resolution check is recorded. Rights: NEEDS_REVIEW; public access verified but redistribution permission not established.