A STUDY ON INVARIANT MEASURES

Author (s) : R.SARANYA, Dr.A.ANITHADEVI, M.VACHALA

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Abstract :
An invariant measure is a measure that is preserved by the dynamics of a system.  More precisely, let (X, B, μ) be a measure space, and let T : X → X be a measurable transformation (i.e., T is a function that preserves the structure of the measure space). Over all, the concept of an invariant measure is a powerful tool in the study of dynamical systems, and has important applications in a wide range of fields, including physics,  biology,  and economics. In the presence of a representation of a compact group  on a banach space E,the following lemma establishes the existence of a family, parametrized by , of positively homogeneous, subadditive functional on E*, each of which is invariant under * and , when restricted to bounded subsets of E*, is continuous with respect to the weak-*topology.Let  be a topological group, E a banach space, and  a representation of  on E. The adjoint representation : ) is a representation of  on E* defined for g  by *(g)  ) for all .Let E be a Banach space and  the banach space of continuous linear operators on E. The composition of two operators in  also belongs to  and clearly, for operators T, S ..An operator in  is invertible if and only if it is one-to-one and onto; the inverse is continuous by the open mapping theorem. Observe that for T, S. A GL(E) is called the general linear group.

Keywords : Invariant, Invertible,Topological group,Banach space.

Conference Date : 13/11/2020

Pages : 102

Paper ID : 20102