Problem 3.3 — Determine the ergodic behavior of the ModΣ-action on the level sets ( iR× R×iR ) ∩κ−1(t) wheret> 2.

Definitions and mathematical statement

For a real parameter $t>2$, define the source's level set \[X_t=(i{\mathbb{R}}\times{\mathbb{R}}\times i{\mathbb{R}})\cap\kappa^{-1}(t),\] where $i{\mathbb{R}}=\{ia:a\in{\mathbb{R}}\}$. The surface $\Sigma$, character-coordinate meaning, function $\kappa$, mapping-class action and measure are introduced only in omitted context. The problem asks to determine the ergodic behavior of the $\operatorname{Mod}_\Sigma$-action on these sets for the entire range $t>2$.

A precise ergodic formulation must first specify an invariant measure $\mu_t$ and the acting group preserving $X_t$. Ergodicity would mean that each invariant measurable subset is null or conull, and a more general behavior description would identify invariant/ergodic pieces and properly discontinuous or wandering regions. The source supplies no assertion that the whole action is ergodic, no almost-everywhere classification, and no measure convention. In character coordinates one must also check preservation of the particular imaginary-real-imaginary slice, rather than only preservation of the polynomial level or of a union of slices.

The source reports that for $t>6$, $X_t$ contains wandering domains corresponding to Fricke spaces of a one-holed Klein bottle. Fricke space here refers to marked hyperbolic structures in the relevant Klein-bottle setting, with boundary/geometric conventions to be specified. The precise meaning of wandering, its possible stabilizer and the relation between $t$ and boundary data are absent. This reported domain is a part of the level set, not an asserted description of its complement or of the ranges $2<t\leq6$. The question therefore includes any remaining measurable dynamics in both parameter regimes; the threshold is kept strictly $t>6$, not $t\geq6$.

Short English statement

Describe the mapping-class dynamics on the specified imaginary-trace level sets for every $t>2$, retaining the known-source wandering-domain qualification only for $t>6$. The action, invariant measure and level function are missing from the excerpt.

Sources

• Local UnsolvedMath registry, permanent ID $11000196$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 3.3, PDF page 222: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction. No separate theorem/reference for the Fricke wandering domains is supplied.