Research classifies linear cocycles using barycenters of measures, indicating connections to ergodic systems.
The barycenters of measures on boundaries of symmetric spaces are applied for classification of linear cocycles over ergodic transformations. We also consider the case of semisimple Lie groups of rank $1$ and recurrence of stationary random walks on them.
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M. E. Lipatov (2026) studied this question.
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