subgroup on its $\taumod$-limit set enjoys the classical convergence property, which is a typical rank one phenomenon.

Regularity, which is a condition on the asymptotic geometry of orbits in the symmetric space, can be converted into an equivalent dynamical condition about a certain contraction behavior of the subgroup on suitable flag manifolds (see Definition \[def:contracting\_sequence\]), allowing one to switch between geometry and dynamics. The contraction behavior here is a higher rank version of the classical convergence (dynamics) property in the theory of Kleinian groups. This yields an equivalent characterization of $\taumod$-regular subgroups as [*$\taumod$-convergence subgroups*]{} (see Definition \[def:conv\]). Also the limit sets, respectively, limit simplices can be defined purely dynamically as the possible limits of contracting sequences in $\Ga$, i.e. of sequences converging to constants on suitable open and dense subsets of the flag manifolds, see Definition \[def:flagl\].

Much of the material in section \[sec:regcontr\] can be found in some form already in the work of Benoist, see [@Benoist §3], in the setting of Zariski dense subgroups of reductive algebraic groups over local fields, notably the notions of [*regularity*]{} and [*contraction*]{}, their essential equivalence, and the notion of [*limit set*]{}. For the sake of completeness we give independent proofs in 