Abstract
For every infinite-type surface with an isolated puncture that admits a shift map and every natural numbern , we study an intrinsically infinite-type mapping classg_(n)and a simple closed curve whose orbit underg_(n)splits intonsubsequences, each of which converges in the coarse Chabauty topology to a geodesic lamination on the surface. These laminations are cyclically permuted byg_(n) , and their union is ag_(n) –invariant geodesic lamination. We further show the attracting limit point ofg_(n)in the boundary of the relative arc graph has weightn . In particular, intrinsically infinite-type loxodromic isometries of the relative arc graph realize every finite weight.