Extension of the lifting and equidistribution framework to capacity with respect to two finite points

Develop a construction of a compact, circled subset E ⊂ C^2 associated to a compact set K ⊂ C with Cantor capacity cap_{a,b}(K)=1 relative to two points a, b ∈ C \ K such that the projection T(z1, z2) = z1/z2 satisfies T(E)=K and the homogeneous capacity cap_h(E)=1; moreover, prove an equidistribution theorem by appropriately lifting univariate polynomials p ∈ Z[z] to polynomial mappings F: C^2 → C^2 so that the normalized zero measures of such mappings on E converge to the Monge–Ampère measure μ_E.

Background

The authors extend Cantor's capacity framework to sets K ⊂ C with respect to 0 and ∞ by lifting to C^2 via the map T(z1, z2)=z1/z2, constructing compact circled sets E with cap_h(E)=1 and showing equidistribution of zeros for suitable polynomial mappings. This enables their main equidistribution results for conjugates of algebraic units.

They note that one can analogously define Cantor capacity cap_{a,b}(K) for points a, b ∈ C \ K using a matrix Ta,b(K). However, they explicitly state that it is unclear how to create the corresponding lift E ⊂ C^2 with T(E)=K and cap_h(E)=1 in this more general setting, and how to prove an equidistribution theorem using their C^2 framework. The obstacle includes devising a suitable lifting of univariate polynomials to polynomial mappings in C^2 aligned with this generalized capacity.