Problem 2.5 — Construct an example of a pseudo-Anosov mapping class for a closed surface which is not ergodic on the SU(2)-characte…

Definitions and mathematical statement

For a closed connected oriented surface $\Sigma_g$, let \[ X_g=\operatorname{Hom}(\pi_1\Sigma_g,\operatorname{SU}(2))/\operatorname{SU}(2) \] be the character quotient by conjugation. In the usual genus-$g\geq2$ setting, give it the finite Goldman symplectic measure from its smooth irreducible locus, normalized to total mass $1$, with the lower-dimensional singular locus treated by the intended convention. An orientation-preserving mapping class $f$ acts by $F_f([\rho])=[\rho\circ(f_*)^{-1}]$, independently of choices up to conjugation, and preserves that measure.

The task is to construct a pseudo-Anosov $f$ on a closed surface for which this transformation is not ergodic. A precise witness is a measurable $A\subset X_g$ such that \[ \mu(F_f^{-1}(A)\mathbin{\triangle}A)=0, \qquad0<\mu(A)<1. \] It is not enough merely to exhibit a periodic character or a measure-zero invariant subset. The source does not explicitly fix the genus, measure, singular-locus or surface-orientation conventions; the displayed statement specifies the standard symplectic-measure interpretation, subject to checking the preceding context.

Short English statement

Construct a closed-surface pseudo-Anosov whose action on the $\operatorname{SU}(2)$-character variety has a proper invariant measurable subset of positive symplectic measure.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Problem 2.5, supplied PDF page 218. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure. No separate reference or individual proposer is supplied for this item.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000192$, exact source code catalogue entry; supplied OPEN-TRIAGE summary reports unverified status.