In this thesis, we study the problem of stationary measure classification, equidistribution and orbit closure classification in three different settings. We use tools from homogeneous dynamics, smooth dynamics and random product of matrices to make progress in each setting. In Chapter 2, we study the problem of classifying stationary measures and orbit closures for non-abelian action on a surface with a given smooth invariant measure. Using a result of Brown and Rodriguez Hertz, we show that under a certain finite verifiable average growth condition, the only nonatomic stationary measure is the given smooth invariant measure, and every orbit closure is either finite or dense. Moreover, every point with infinite orbit equidistributes on the surface with respect to the smooth invariant measure. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the homogeneous setting, and the result of Eskin-Mirzakhani in the setting of moduli spaces of translation surfaces. We then apply this result to two concrete settings, namely discrete perturbation of the standard map and Out(F_2)-action on a certain character variety. We verify the growth condition analytically in the former setting, and verify numerically in the latter setting. In Chapter 3, we provide a self-contained proof of the classification of stationary measures for linear actions on vector spaces. This will be a major input of the result in the next chapter. In Chapter 4, we study the problem of classifying stationary measures on homogeneous spaces of the form G/H, where G is a connected real Lie group, and H is a closed unimodular subgroup of G. Under an assumption of relative uniform expansion, we show that the stationary measures can be decomposed into homogeneous parts and generalized Bernoulli convolutions. The main tools used are a relative version of the technique of Eskin-Lindenstrauss, and the measure classification result of linear action on real vector spaces from Chapter 3.




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