Limit sets of Teichm\"uller geodesics with minimal nonuniquely ergodic vertical foliation, II

TL;DR

This paper characterizes the limit sets of Teichmüller geodesics with nonuniquely ergodic vertical foliations, providing conditions for convergence to complex laminations and constructing examples with maximal measure space dimension.

Contribution

It introduces conditions ensuring certain limit behaviors of Teichmüller geodesics and constructs examples with maximal measure space dimension on closed surfaces.

Findings

Sequences form infinite quasi-geodesics in the curve complex

Limit in Gromov boundary is a nonuniquely ergodic lamination

Constructed sequences with maximal measure space dimension

Abstract

Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus $g$ for which the space of measures has the maximal dimension $3g-3$, for example. We also study the limit sets in the Thurston boundary of Teichm\"uller geodesic rays defined by quadratic differentials whose vertical foliations are obtained from the constructions mentioned above. We prove that such examples exist for which the limit is a cycle in the $1$-skeleton of the simplex of projective classes of measures visiting every vertex.