Problem 3 — Give a useful (piecewise) tropical description of the two elementary transformations.

Definitions and mathematical statement

Let the two elementary transformations and their coordinate domains be those defined before this problem; their formulas and geometric meanings are not reproduced in the excerpt. The request is to express both transformations usefully by piecewise tropical formulas, and thereby obtain the stated piecewise tropical-polynomial mapping-class-group representation. The formulas must reproduce the original coordinate changes on their admissible domains and respect the move relations needed to descend to the mapping class group; no faithfulness is expressly required.

For orientation about terminology, in a max-plus convention tropical addition and multiplication are $a\oplus b=\max(a,b)$ and $a\odot b=a+b$. A tropical polynomial is then a finite maximum of affine-linear forms with nonnegative integral exponent vectors; allowing negative exponents gives Laurent expressions. A min-plus convention is also possible. The source specifies neither convention nor allowed subtraction/Laurent operations or chamber boundaries, so this reference definition does not choose the missing formula class. The chart domains may have constraints and need not be a full Euclidean space.

The source asks what properties the resulting representation has, particularly under iteration of a mapping class. Once its coordinate action $T_\varphi$ is specified, this means analyzing $T_\varphi^n$ and its relation to iteration on the geometric objects. No particular growth law, invariant or dynamical classification is asserted in the excerpt.

An alternative suggested coordinate system, attributed to [9], uses intersection numbers with a family of curves $(c_j)_{j\in J}$. For a compactly supported measured foliation $\mathcal F$, it records \[I(\mathcal F)=\left(i(\mathcal F,c_j)\right)_{j\in J},\] where $i$ is geometric measured intersection number. The source says these numbers coordinatize the foliations, subject to relations, and tentatively suggests that Theorem 3 could compute them. The curve family, relations and Theorem 3 are absent, so no assertion of effective recovery or unrestricted independent coordinates is inserted. The final request for further applications/consequences is open-ended and supplies no additional fixed theorem.

Short English statement

Give tropical formulas for two specified but omitted coordinate transformations, study the induced mapping-class action under iteration, and explore intersection-number coordinates and further applications. The source's tentative computational suggestion remains tentative.

Sources

• Local UnsolvedMath registry, permanent ID $11000133$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 3, PDF page 117: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction.
• Internal reference [9] and Theorem 3 provide the alternative coordinates and proposed computation, but their content/full citation is not supplied. The summary only mentions unspecified tropical Teichmüller literature.