Stationary Measures on Homogeneous Spaces

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Rajabzadeh, Hesameddin | 2016

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 48865 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Haji Mirsadeghi, Mir Omid; Nasiri, Meysam
Abstract:
Let be a sub-semigroup of SL(d;Z) which acts strongly irreducible on Rd, i.e. there is not any invariant finite union of nontrivial proper subspaces of Rd. acts on Td by homeomorphisms. We will show that under these assumptions every infinite invariant subset of Td under the action of is dense in Td. For instance, if x is a point in Td which is not rational, then x = Td. In this thesis, we will follow the method of Y. Benoist and J.F. Quint for proving such results. Their method is to prove some kind of rigidity for the stationary measures for the action of on Td. Actually, every atomless stationary measure is the lebesgue measure on Td which is invariant under all the elements of SL(d;Z)
Keywords:
Homogeneous Space ; Stationary Measure ; Matrices Random Product ; Semisimple Lie Group ; Strongly Irreducible Action